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Finished all chapters and definitions. I need to add subsections and see if there's any theorems or definitions in the appendicies that are worth adding to this as well.

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chapter-6/bilinear-and-quadratic-forms.tex
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\section{Bilinear and Quadratic Forms}
\begin{definition}
\hfill\\
Let $V$ be a vector space over a field $\F$. A function $H$ from the set $V \times V$ if ordered pairs of vectors to $\F$ is called a \textbf{bilinear form} on $V$ if $H$ is linear in each variable when the other variable is held fixed; that is, $H$ is a bilinear form on $V$ if
\begin{enumerate}
\item $H(ax_1 + x_2, y) = aH(x_1,y) + H(x_2,y)$ for all $x_1, x_2, y \in V$ and $a \in \F$.
\item $H(x,ay_1+y_2) = aH(x,y_1) + H(x,y_2)$ for all $x,y_1,y_2 \in V$ and $a \in \F$.
\end{enumerate}
\end{definition}
\begin{notation}
\hfill\\
We denote the set of all bilinear forms on $V$ by $\mathcal{B}(V)$.
\end{notation}
\begin{definition}
\hfill\\
Let $V$ be a vector space, let $H_1$ and $H_2$ be bilinear forms on $V$, and let $a$ be a scalar. We define the \textbf{sum} $H_1 + H_2$ and the \textbf{scalar product} $aH_1$ by the equations
\[(H_1 + H_2)(x,y) = H_1(x,y) + H_2(x,y)\]
and
\[(aH_1)(x,y) = a(H_1(x,y))\ \ \ \text{for all}\ x,y \in V.\]
\end{definition}
\begin{theorem}
\hfill\\
For any vector space $V$, the sum of two bilinear forms and the product of a scalar and a bilinear form on $V$ are again bilinear forms on $V$. Furthermore, $\mathcal{B}(V)$ is a vector space with respect to these operations.
\end{theorem}
\begin{definition}
let $\beta = \{v_1, v_2, \dots, v_n\}$ be an ordered basis for an $n$-dimensional vector space $V$, and let $H \in \mathcal{B}(V)$. We can associate $H$ with an $n \times n$ matrix $A$ whose entry in row $i$ and column $j$ is defined by
\[A_{ij} = H(v_i, v_j)\ \ \ \ \text{for}\ i,j = 1, 2, \dots, n.\]
The matrix $A$ is called the \textbf{matrix representation} of $H$ with respect to the ordered basis $\beta$ and is denoted by $\psi_\beta(H)$.
\end{definition}
\begin{theorem}
\hfill\\
For any $n$-dimensional vector space $V$ over $\F$ and any ordered basis $\beta$ for $V$, $\psi_\beta : \mathcal{B}(V) \to M_{n \times n}(\F)$ is an isomorphism.
\end{theorem}
\begin{corollary}
\hfill\\
For any $n$-dimensional vector space $V$, $\mathcal{B}(V)$ has dimension $n^2$.
\end{corollary}
\begin{corollary}
\hfill\\
Let $V$ be an $n$-dimensional vector space over $\F$ with ordered basis $\beta$. If $H \in \mathcal{B}(V)$ and $A \in M_{n \times n}(\F)$, then $\psi_\beta(H) = A$ if and only if $H(x,y) = [\phi_\beta(x)]^tA[\phi_\beta(y)]$ for all $x,y \in V$.
\end{corollary}
\begin{corollary}
\hfill\\
Let $\F$ be a field, $n$ a positive integer, and $\beta$ be the standard ordered basis for $F^n$. Then for any $H \in \mathcal{B}(F^n)$, there exists a unique square matrix $A \in M_{n \times n}(\F)$, names, $A = \psi_\beta(H)$, such that
\[H(x,y) = x^tAy\ \ \ \ \text{for all}\ x,y \in F^n.\]
\end{corollary}
\begin{definition}
\hfill\\
Let $A,B \in M_{n \times n}(\F)$. Then $B$ is said to be \textbf{congruent} to $A$ if there exists an invertible matrix $Q \in M_{n \times n}(\F)$ such that $B = Q^tAQ$.
\end{definition}
\begin{theorem}
\hfill\\
Let $V$ be a finite-dimensional vector space with ordered bases $\beta = \{v_1, v_2, \dots, v_n\}$ and $\gamma = \{w_1, w_2, \dots, w_n\}$, and let $Q$ be the change of coordinate matrix changing $\gamma$-coordinates into $\beta$-coordinates. Then, for any $H \in \mathcal{B}(V)$, we have $\psi_\gamma(H) = Q^t\psi_\beta(H)Q$. Therefore $\psi_\gamma(H)$ is congruent to $\psi_\beta(H)$.
\end{theorem}
\begin{corollary}
\hfill\\
Let $V$ be an $n$-dimensional vector space with ordered basis $\beta$, and let $H$ be a bilinear form on $V$. For any $n \times n$ matrix $B$, if $B$ is congruent to $\psi_\beta(H)$, then there exists an ordered basis $\gamma$ for $V$ such that $\psi_\gamma(H) = B$. Furthermore, if $B = Q^t\psi_\beta(H)Q$ for some invertible matrix $Q$, then $Q$ changes $\gamma$-coordinates into $\beta$-coordinates.
\end{corollary}
\begin{definition}
\hfill\\
A bilinear form $H$ on a vector space $V$ is \textbf{symmetric} if $H(x,y) = H(y,x)$ for all $x,y \in V$.
\end{definition}
\begin{theorem}
\hfill\\
Let $H$ be a bilinear form on a finite-dimensional vector space $V$, and let $\beta$ be an ordered basis for $V$. Then $H$ is symmetric if and only if $\psi_\beta(H)$ is symmetric.
\end{theorem}
\begin{definition}
\hfill\\
A bilinear form $H$ on a finite-dimensional vector space $V$ is called \textbf{diagonalizable} if there is an ordered basis $\beta$ for $V$ such that $\psi_\beta(H)$ is a diagonal matrix.
\end{definition}
\begin{corollary}
\hfill\\
Let $H$ be a diagonalizable bilinear form on a finite-dimensional vector space $V$. Then $H$ is symmetric.
\end{corollary}
\begin{definition}
\hfill\\
The \textbf{characteristic} of a field $\F$ is the smallest positive integer $p$ for which a sum of $p$ 1's equals $0$. If no such positive integer exists, then $\F$ is said to have \textbf{characteristic zero}.
\end{definition}
\begin{lemma}
\hfill\\
Let $H$ be a nonzero symmetric bilinear form on a vector space $V$ over a field $\F$ not of characteristic two. Then there is a vector $x$ in $V$ such that $H(x,x) \neq 0$.
\end{lemma}
\begin{theorem}
\hfill\\
Let $V$ be a finite-dimensional vector space over a field $\F$ not of characteristic two. Then every symmetric bilinear form on $V$ is diagonalizable.
\end{theorem}
\begin{corollary}
\hfill\\
Let $\F$ be a field that is not of characteristic two. If $A \in M_{n \times n}(\F)$ is a symmetric matrix, then $A$ is congruent to a diagonal matrix.
\end{corollary}
\begin{definition}
\hfill\\
Let $V$ be a vector space over $\F$. A function $K: V \to \F$ is called a \textbf{quadratic form} if there exists a symmetric bilinear form $H \in \mathcal{B}(V)$ such that
\[K(x) = H(x, x)\ \ \ \text{for all}\ x \in V.\]
\end{definition}
\begin{theorem}
\hfill\\
Let $V$ be a finite-dimensional real inner product space, and let $H$ be a symmetric bilinear form on $V$. Then there exists an orthonormal basis $\beta$ for $V$ such that $\psi_\beta(H)$ is a diagonal matrix.
\end{theorem}
\begin{corollary}
\hfill\\
Let $K$ be a quadratic form on a finite-dimensional real inner product space $V$. There exists an orthonormal basis $\beta = \{v_1, v_2, \dots, v_n\}$ for $V$ and scalars $\lambda_1, \lambda_2, \dots, \lambda_n$ (not necessarily distinct) such that if $x \in V$ and
\[x = \sum_{i=1}^{n}s_iv_i,\ \ \ s_i \in \R\]
then
\[K(x) = \sum_{i=1}^{n}\lambda_is_i^2.\]
In fact, if $H$ is the symmetric bilinear form determined by $K$, then $\beta$ can be chosen to be any orthonormal basis for $V$ such that $\psi_\beta(H)$ is a diagonal matrix.
\end{corollary}
\begin{definition}
\hfill\\
Let $z=f(t_1, t_2, \dots, t_n)$ be a fixed real-valued function of $n$ real variables for which all third-order partial derivatives exist and are continuous. The function $f$ is said to have a \textbf{local maximum} at point $p \in \R^n$ if there exists a $\delta > 0$ such that $f(p) \geq f(x)$ whenever $||x - p|| < \delta$. Likewise, $f$ has a \textbf{local minimum} at $p \in \R^n$ if there exists a $\delta > 0$ such that $f(p) \leq f(x)$ whenever $||x - p|| < \delta$. If $f$ has either a local minimum or a local maximum at $p$, we say that $f$ has a \textbf{local extremum} at $p$. A point $p \in \R^n$ is called a \textbf{critical point} of $f$ if $\displaystyle\frac{\partial f(p)}{\partial(t_i)} = 0$ for $i = 1, 2, \dots, n$. It is a well known fact that if $f$ has a local extremum at a point $p \in \R^n$, then $p$ is a critical point of $f$. For, if $f$ has a local extremum at $p=(p_1, p_2, \dots, p_n)$, then for any $i = 1, 2, \dots, n$, the function $\phi_i$ defined by $\phi_i(t) = f(p_1, p_2, \dots, p_{i-1}, t, p_{i+1}, \dots, p_n)$ has a local extremum at $t = p_i$. So, by an elementary single-variable argument,
\[\frac{\partial f(p)}{\partial t_i} = \frac{d\phi_i(p_i)}{dt} = 0.\]
Thus $p$ is a critical point of $f$. But critical points are not necessarily local extrema.
The second-order partial derivatives of $f$ at a critical point $p$ can often be used to test for local extremum at $p$. These partials determine a matrix $A(p)$ in which the row $i$, column $j$ entry is
\[\frac{\partial^2f(p)}{(\partial t_i)(\partial t_j)}.\]
This matrix is called the \textbf{Hessian matrix} of $f$ at $p$. Note that if the third order partial derivatives of $f$ are continuous, then the mixed second-order partials of $f$ at $p$ are independent of the order in which they are taken, and hence $A(p)$ is a symmetric matrix. In this case, all of the eigenvalues of $A(p)$ are real.
\end{definition}
\begin{theorem}[\textbf{The Second Derivative Test}]
\hfill\\
Let $f(t_1, t_2, \dots, t_n)$ be a real-valued function in $n$ real variables for which all third-order partial derivatives exist and are continuous. Let $p = (p_1, p_2, \dots, p_n)$ be a critical point of $f$, and let $A(p)$ be the Hessian of $f$ at $p$.
\begin{enumerate}
\item If all eigenvalues of $A(p)$ are positive, then $f$ has a local minimum at $p$.
\item If all eigenvalues of $A(p)$ are negative, then $f$ has a local maximum at $p$.
\item If $A(p)$ has at least one positive and at least one negative eigenvalue, then $f$ has no local extremum at $p$ ($P$ is called a \textbf{saddle-point} of $f$).
\item If $\rank{A(p)} < n$ and $A(p)$ does not have both positive and negative eigenvalues, then the second derivative test is inconclusive.
\end{enumerate}
\end{theorem}
\begin{definition}
\hfill\\
The \textbf{rank} of a bilinear form is the rank of any of its matrix representations.
\end{definition}
\begin{theorem}[\textbf{Sylvester's Law of Inertia}]
Let $H$ be a symmetric bilinear form on a finite-dimensional real vector space $V$. Then the number of positive diagonal entries and the number of negative diagonal entries in any diagonal matrix representation of $H$ are each independent of the diagonal representation.
\end{theorem}
\begin{definition}
\hfill\\
The number of positive diagonal entries in a diagonal representation of a symmetric bilinear form on a real vector space is called the \textbf{index} of the form. The difference between the number of positive and the number of negative diagonal entries in a diagonal representation of a symmetric bilinear form is called the \textbf{signature} of the form. The three terms \textit{rank}, \textit{index}, and \textit{signature} are called \textbf{invariants} of the bilinear form because they are invariant with respect to matrix representations. These same terms apply to the associated quadratic form. Notice that the values of any two of these invariants determine the value of the third.
\end{definition}
\begin{corollary}[\textbf{Sylvester's Law of Inertia for Matrices}]
\hfill\\
Let $A$ be a real symmetric matrix. Then the number of positive diagonal entries and the number of negative diagonal entries in any diagonal matrix congruent to $A$ is independent of the choice of the diagonal matrix.
\end{corollary}
\begin{definition}
\hfill\\
Let $A$ be a real symmetric matrix, and let $D$ be a diagonal matrix that is congruent to $A$. The number of positive diagonal entries of $D$ is called the \textbf{index} of $A$. The difference between the number of positive diagonal entries and the number of negative diagonal entries of $D$ is called the \textbf{signature} of $A$. As before, the \textit{rank}, \textit{index}, and \textit{signature} of a matrix are called the \textbf{invariants} of the matrix, and the values of any two of these invariants determine the value of the third.
\end{definition}
\begin{corollary}
\hfill\\
Two real symmetric $n \times n$ matrices are congruent if and only if they have the same invariants.
\end{corollary}
\begin{corollary}
\hfill\\
A real symmetric $n \times n$ matrix $A$ has index $p$ and rank $r$ if and only if $A$ is congruent to $J_{pr}$, defined by
\[J_{pr} = \begin{pmatrix}
I_p & O & O \\
O & -I_{r-p} & O \\
O & O & O
\end{pmatrix}\]
\end{corollary}
\begin{definition}
\hfill\\
For a $2 \times 2$ matrix $M$ with columns $x$ and $y$, the bilinear form $H(M)=H(x,y)$ is called the \textbf{permanent} of $M$.
\end{definition}
chapter-6/conditioning-and-the-rayleigh-quotient.tex
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\section{Conditioning and the Rayleigh Quotient}
\begin{definition}
\hfill\\
For a system of linear equations $Ax = b$, two types of errors must be considered for real-world application. First, experimental errors arise in the collection of data since no instruments can provide completely accurate measurements. Second, computers introduce round-off errors. One might intuitively feel that small relative changes in the coefficients of the system cause small relative errors in the solution. A system that has this property is called \textbf{well-conditioned}; otherwise, the system is called \textbf{ill-conditioned}.
\end{definition}
\begin{notation}
\hfill\\
Let $Ax = b$ be a system of linear equations. We use the notation $\delta b$ to denote the vector $b' - b$, where $b$ is the vector in the original system, and $b'$ is the vector in the modified system.
\end{notation}
\begin{definition}
\hfill\\
Let $Ax = b$ be a system of linear equations. We define the \textbf{relative change} in a vector $b$ to be the scalar $||\delta b||/ ||b||$, where $|| \cdot ||$ denotes the standard norm on $\C^n$ (or $\R^n$); that is, $||b|| = \sqrt{\lr{b, b}}$. Similar definitions hold for the \textbf{relative change} of $x$.
\end{definition}
\begin{definition}
Let $A$ be a complex (or real) $n \times n$ matrix. Define the \textbf{(Euclidean) norm} of $A$ by
\[||A|| = \max_{x \neq 0} \frac{||Ax||}{||x||},\]
where $x \in \C^n$ or $x \in \R^n$.\\
Intuitively, $||A||$ represents the maximum \textit{magnification} of a vector by the matrix $A$.
\end{definition}
\begin{definition}
Let $B$ be an $n \times n$ self-adjoint matrix. The \textbf{Rayleigh quotient} for $x \neq 0$ is defined to be the scalar $R(x) = \lr{Bx, x}/ ||x||^2$.
\end{definition}
\begin{theorem}
\hfill\\
For a self-adjoint matrix $B \in M_{n \times n}(\F)$, we have that $\displaystyle\max_{x \neq 0}R(x)$ is the largest eigenvalue of $B$ and $\displaystyle\min_{x \neq 0}R(x)$ is the smallest eigenvalue of $B$.
\end{theorem}
\begin{corollary}
\hfill\\
For any square matrix $A$, $||A||$ is finite and, in fact, equals $\sqrt{\lambda}$, where $\lambda$ is the largest eigenvalue of $A^*A$.
\end{corollary}
\begin{lemma}
\hfill\\
For any square matrix $A$, $\lambda$ is an eigenvalue of $A^*A$ if and only if $\lambda$ is an eigenvalue of $AA^*$.
\end{lemma}
\begin{corollary}
\hfill\\
Let $A$ be an invertible matrix. Then $||A^{-1}|| = 1/\sqrt{\lambda}$, where $\lambda$ is the smallest eigenvalue of $A^*A$.
\end{corollary}
\begin{definition}
\hfill\\
Let $A$ be an invertible matrix. The number $||A||\cdot||A^{-1}||$ is called the \textbf{condition number} of $A$ and is denoted $\text{cond}(A)$.
\end{definition}
\begin{theorem}
\hfill\\
For the system $Ax = b$, where $A$ is invertible and $b \neq 0$, the following statements are true.
\begin{enumerate}
\item For any norm, $||\cdot||$, we have $\displaystyle\frac{1}{\text{cond}(A)}\frac{||\delta b ||}{||b||} \leq \frac{||\delta x||}{||x||} \leq \text{cond}(A)\frac{||\delta b||}{||b||}$.
\item If $||\cdot||$ is the Euclidean norm, then $\text{cond}(A) = \sqrt{\lambda_1/\lambda_n}$, where $\lambda_1$ and $\lambda_n$ are the largest and smallest eigenvalues, respectively, of $A^*A$.
\end{enumerate}
\end{theorem}
chapter-6/einsteins-special-theory-of-relativity.tex
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\section{Einstein's Special Theory of Relativity}
\begin{definition}[\textbf{Axioms of the Special Theory of Relativity}]
\hfill\\
The basic problem is to compare two different inertial (non-accelerating) coordinate systems $S$ and $S'$ in three-space ($\R^3$) that are in motion relative to each other under the assumption that the speed of light is the same when measured in either system. We assume that $S'$ moves at a constant velocity in relation to $S$ as measured from $S$. To simplify matters, let us suppose that the following conditions hold:
\begin{enumerate}
\item The corresponding axes of $S$ and $S'$ ($x$ and $x'$, $y$ and $y'$, $z$ and $z'$) are parallel, and the origin of $S'$ moves in the positive direction of the $x$-axis of $S$ at a constant velocity $v > 0$ relative to $S$.
\item Two clocks $C$ and $C'$ are placed in space - the first stationary relative to the coordinate system $S$ and the second stationary relative to the coordinate system $S'$. These clocks are designed to give real numbers in units of seconds as readings. The clocks are calibrated so that at the instant the origins of $S$ and $S'$ coincide, both clocks give the reading zero.
\item The unit of length is the \textbf{light second} (the distance light travels in 1 second), and the unit of time is the second. Note that, with respect to these units, the speed of light is 1 light second per second.
\end{enumerate}
Given any event (something whose position and time of occurrence can be described), we may assign a set of \textit{space-time coordinates} to it. For example, if $p$ is an event that occurs at position
\[\begin{pmatrix} x \\ y \\ z \end{pmatrix}\]
relative to $S$ and at time $t$ as read on clock $C$, we can assign to $p$ the set of coordinates
\[\begin{pmatrix} x \\ y \\ z \\ t \end{pmatrix}.\]
This ordered 4-tuple is called the \textbf{space-time coordinates} of $p$ relative to $S$ and $C$. Likewise, $p$ has a set of space-time coordinates
\[\begin{pmatrix} x' \\ y' \\ z' \\ t' \end{pmatrix}\]
relative to $S'$ and $C'$.
For a fixed velocity $v$, let $T_v: \R^4 \to \R^4$ be the mapping defined by
\[T_v \begin{pmatrix}x \\ y \\ z \\ t \end{pmatrix} = \begin{pmatrix} x' \\ y' \\ z' \\ t' \end{pmatrix},\]
where
\[\begin{pmatrix}x \\ y \\ z \\ t \end{pmatrix}\ \ \ \text{and}\ \ \ \begin{pmatrix} x' \\ y' \\ z' \\ t' \end{pmatrix}\]
are the space-time coordinates of the same event with respect to $S$ and $C$ and with respect to $S'$ and $C'$, respectively.
Einstein made certain assumptions about $T_v$ that led to his special theory of relativity. We formulate an equivalent set of assumptions.
\begin{enumerate}
\item[(R 1)] The speed of any light beam, when measured in either coordinate system using a clock stationary relative to that coordinate system, is 1.
\item[(R 2)] The mapping $T_v: \R^4 \to \R^4$ is an isomorphism.
\item[(R 3)] If
\[T_v\begin{pmatrix}
x \\ y \\ z \\ t
\end{pmatrix} = \begin{pmatrix}
x' \\ y' \\ z' \\ t'
\end{pmatrix}\]
then $y' = y$ and $z' = z$.
\item[(R 4)] If
\[T_v\begin{pmatrix}
x \\ y_1 \\ z_1 \\ t
\end{pmatrix} = \begin{pmatrix}
x' \\ y' \\ z' \\ t'
\end{pmatrix}\ \ \ \ \text{and}\ \ \ \ T_v\begin{pmatrix}
x \\ y_2 \\ z_2 \\ t
\end{pmatrix} = \begin{pmatrix}
x'' \\ y'' \\ z'' \\ t''
\end{pmatrix}\]
then $x'' = x'$ and $t'' = t'$.
\item[(R 5)] The origin of $S$ moves in the negative direction of the $x'$-axis of $S'$ at the constant velocity $-v < 0$ as measured from $S'$.
\end{enumerate}
Axioms (R 3) and (R 4) tell us that for $p \in \R^4$, the second and third coordinates of $T_v(p)$ are unchanged and the first and fourth coordinates of $T_v(p)$ are independent of the second and third coordinates of $p$.
These five axioms completely characterize $T_v$. The operator $T_v$ is called the \textbf{Lorentz transformation} in direction $x$.
\end{definition}
\begin{theorem}
\hfill\\
On $\R^4$, the following statements are true.
\begin{enumerate}
\item $T_v(e_i) = e_i$ for $i = 2,3$.
\item $\lspan{\{e_2, e_3\}}$ is $T_v$-invariant.
\item $\lspan{\{e_1, e_4\}}$ is $T_v$-invariant.
\item Both $\lspan{\{e_2, e_3\}}$ and $\lspan{\{e_1, e_4\}}$ are $T_v^*$-invariant.
\item $T_v^*(e_i) = e_i$ for $i=2,3$.
\end{enumerate}
\end{theorem}
\begin{theorem}
\hfill\\
If $\lr{L_A(w),w} = 0$ for some $w \in \R^4$, then
\[\lr{T_v^*L_AT_V(w), w} = 0.\]
\end{theorem}
\begin{theorem}
\hfill\\
There exist nonzero scalars $a$ and $b$ such that
\begin{enumerate}
\item $T_v^*L_AT_v(w_1) = aw_2$.
\item $T_v^*L_AT_v(w_2) = bw_1$.
\end{enumerate}
\end{theorem}
\begin{corollary}
\hfill\\
Let $B_v = [T_v]_\beta$, where $\beta$ is the standard ordered basis for $\R^4$. Then
\begin{enumerate}
\item $B_v^*AB_v = A$.
\item $T_v^*L_AT_v = L_A$.
\end{enumerate}
\end{corollary}
\begin{theorem}
\hfill\\
Let $\beta$ be the standard ordered basis for $\R^4$. Then
\[[T_V]_\beta = B_v = \begin{pmatrix}
\frac{1}{\sqrt{1 - v^2}} & 0 & 0 & \frac{-v}{\sqrt{1 - v^2}} \\
0 & 1 & 0 & 0 \\
0 & 0 & 1 & 0 \\
\frac{-v}{\sqrt{1 - v^2}} & 0 & 0 & \frac{1}{\sqrt{1 - v^2}}
\end{pmatrix}\]
\end{theorem}
chapter-6/inner-products-and-norms.tex
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\section{Inner Products and Norms}
\begin{definition}
\hfill\\
Let $V$ be a vector space over $\F$. An \textbf{inner product} on $V$ is a function that assigns, to every ordered pair of vectors $x$ and $y$ in $V$, a scalar in $\F$, denoted by $\lr{x, y}$, such that for all $x, y$ and $z$ in $V$ and all $c$ in $\F$, the following hold:
\begin{enumerate}
\item $\lr{x + z, y} = \lr{x, y} + \lr{z,y}$.
\item $\lr{cx, y} = c\lr{x,y}$.
\item $\overline{\lr{x,y}} = \lr{y,x}$, where the bar denoted complex conjugation.
\item $\lr{x, x} > 0$ if $x \neq 0$.
\end{enumerate}
\end{definition}
\begin{definition}\label{Definition 6.2}
\hfill\\
If $x$ and $y$ are in the vector space $\C^n$, define their inner product to be
\[\lr{x,y} = \sum_{k=1}^{n}x_k\overline{y_k}.\]
This is called the (complex) \textbf{standard inner product}.\\
When $\F = \R$ the conjugations are not needed, and in early courses this standard inner product is usually called the \textit{dot product} and is denoted by $x \cdot y$ instead of $\lr{x, y}$.
\end{definition}
\begin{definition}\label{Conjugate Transpose}
\hfill\\
Let $A \in M_{m \times n}(\F)$. We define the \textbf{conjugate transpose} or \textbf{adjoint} of $A$ to be the $n \times m$ matrix $A^*$ such that $(A^*)_{ij} = \overline(A_{ij})$ for all $i,j$.
\end{definition}
\begin{definition}\label{Definition 6.4}
\hfill\\
Given two complex number-valued $n \times m$ matrices $A$, and $B$, written explicitly as
\[A = \begin{pmatrix}
A_{11} & A_{12} & \dots & A_{1m} \\
A_{21} & A_{22} & \dots & A_{2m} \\
\vdots & \vdots & \ddots & \vdots \\
A_{n1} & A_{n2} & \dots & A_{nm}
\end{pmatrix},\ \ \ \ \begin{pmatrix}
B_{11} & B_{12} & \dots & B_{1m} \\
B_{21} & B_{22} & \dots & B_{2m} \\
\vdots & \vdots & \ddots & \vdots \\
B_{n1} & B_{n2} & \dots & B_{nm}
\end{pmatrix}\]
the \textbf{Frobenius inner product} is defined as,
\[\lr{A, B}_\F = \sum_{i,j}\overline{A_{ij}}B_{ij} = \text{tr}(A^*A)\]
Where the overline denotes the complex conjugate, and $A^*$ denotes the conjugate transpose (\autoref{Conjugate Transpose}). Explicitly this sum is
\[\begin{aligned}
\lr{A,B}_\F = & \overline{A_{11}}B_{11} + \overline{A_{12}}B_{12} + \dots + \overline{A_{1m}}B_{1m} \\
& + \overline{A_{21}}B_{21} + \overline{A_{22}}B_{22} + \dots + \overline{A_{2m}}B_{2m} \\
& \vdots \\
& + \overline{A_{n1}}B_{n1} + \overline{A_{n2}}B_{n2} + \dots + \overline{A_{nm}}B_{nm} \\
\end{aligned}\]
\end{definition}
\begin{definition}
\hfill\\
A vector space $V$ over $\F$ endowed with a specific inner product is called an \textbf{inner product space}. If $\F=\C$, we call $V$ a \textbf{complex inner product space}, whereas if $\F = \R$, we call $V$ a \textbf{real inner product space}.
\end{definition}
\begin{notation}
\hfill\\
For the remainder of this chapter, $F^n$ denotes the inner product space with the standard inner product as defined in \autoref{Definition 6.2}. Likewise, $M_{n \times n}(F)$ denotes the inner product space with the Frobenius inner product as defined in \autoref{Definition 6.4}.
\end{notation}
\begin{theorem}\label{Theorem 6.1}
\hfill\\
Let $V$ be an inner product space. Then for $x,y,z \in V$ and $c \in \F$ the following statements are true.
\begin{enumerate}
\item $\lr{x, y+z} = \lr{x,y} + \lr{x,z}$.
\item $\lr{x,cy} = \overline{c}\lr{x,y}$.
\item $\lr{x,0} = \lr{0,x} = 0$.
\item $\lr{x,x} = 0$ if and only if $x = 0$.
\item If $\lr{x,y} = \lr{x,z}$ for all $x \in V$, then $y=z$.
\end{enumerate}
\end{theorem}
\begin{remark}
\hfill\\
It should be observed that the (1) and (2) of \autoref{Theorem 6.1} show that the inner product is \textbf{conjugate linear} in the second component.
\end{remark}
\begin{definition}
\hfill\\
Let $V$ be an inner product space. For $x \in V$, we define the \textbf{norm} or \textbf{length} of $x$ by\\ $||x|| = \sqrt{\lr{x,x}}$.
\end{definition}
\begin{theorem}
\hfill\\
Let $V$ be an inner product space over $\F$. Then for all $x,y \in V$ and $c \in \F$, the following statements are true.
\begin{enumerate}
\item $||cx|| = |c|\cdot||x||$.
\item $||x|| = 0$ if and only if $x = 0$. In any case, $||x|| \geq 0$.
\item (Cauchy-Schwarz Inequality) $|\lr{x,y}| \leq ||x||\cdot||y||$.
\item (Triangle Inequality) $||x + y|| \leq ||x|| + ||y||$.
\end{enumerate}
\end{theorem}
\begin{definition}
\hfill\\
Let $V$ be an inner product space. Vectors $x$ and $y$ in $V$ are \textbf{orthogonal} (\textbf{perpendicular}) if $\lr{x,y} = 0$. A subset $S$ of $V$ is \textbf{orthogonal} if any two distinct vectors in $S$ are orthogonal. A vector $x$ in $V$ is a \textbf{unit vector} if $||x|| = 1$. Finally, a subset $S$ of $V$ is \textbf{orthonormal} if $S$ is orthogonal and consists entirely of unit vectors.\\
Note that if $S = \{v_1, v_2, \dots\}$, then $S$ is orthonormal if and only if $\lr{v_i, v_j} = \delta_{ij}$, where $\delta_{ij}$ denotes the Kronecker delta. Also, observe that multiplying vectors by nonzero scalars does not affect their orthogonality and that if $x$ is any nonzero vector, then $(1/||x||)x$ is a unit vector. The process of multiplying a nonzero vector by the reciprocal of its length is called \textbf{normalizing}.
\end{definition}
\begin{lemma}[\textbf{Parallelogram Law}]
\hfill\\
Let $V$ be an inner product space. Then
\[||x + y||^2 + ||x - y||^2 = 2||x||^2 + 2||y||^2\ \ \text{for all}\ x,y \in V\]
\end{lemma}
\begin{definition}
\hfill\\
Let $V$ be a vector space $\F$, where $\F$ is either $\R$ of $\C$. Regardless of whether $V$ is or is not an inner product space, we may still define a norm $||\cdot||$ as a real-values function on $V$ satisfying the following three conditions for all $x,y \in V$ and $a \in \F$:
\begin{enumerate}
\item $||x|| \geq 0$, and $||x|| = 0$ if and only if $x = 0$.
\item $||ax|| = |a| \cdot ||x||$.
\item $||x + y|| \leq ||x|| + ||y||$.
\end{enumerate}
\end{definition}
\begin{definition}
\hfill\\
Let $||\cdot||$ be a norm on a vector space $V$, and define, for each ordered pair of vectors, the scalar $d(x, y) = ||x - y||$. This is called the \textbf{distance} between $x$ and $y$.
\end{definition}
chapter-6/normal-and-self-adjoint-operators.tex
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\section{Normal and Self-Adjoing Operators}
\section{Normal and Self-Adjoint Operators}
\begin{lemma}
\hfill\\
Let $T$ be a linear operator on a finite-dimensional inner product space $V$. If $T$ has an eigenvector, then so does $T^*$.
\end{lemma}
\begin{theorem}[\textbf{Schur}]
\hfill\\
Let $T$ be a linear operator on a finite-dimensional inner product space $V$. Suppose that the characteristic polynomial of $T$ splits. Then there exists an orthonormal basis $\beta$ for $V$ such that the matrix $[T]_\beta$ is upper triangular.
\end{theorem}
\begin{definition}
\hfill\\
Let $V$ be an inner product space, and let $T$ be a linear operator on $V$. We say that $T$ is \textbf{normal} if $TT^* = T^*T$. An $n \times n$ real or complex matrix $A$ is \textbf{normal} if $AA^* = A^*A$.
\end{definition}
\begin{theorem}
\hfill\\
Let $V$ be an inner product space, and let $T$ be a normal operator on $V$. Then the following statements are true.
\begin{enumerate}
\item $||T(x)|| = ||T^*(x)||$ for all $x \in V$.
\item $T - cI$ is normal for every $c \in \F$.
\item If $x$ is an eigenvector of $T$, then $x$ is also an eigenvector of $T^*$. In fact, if $T(x) = \lambda x$, then $T^*(x) = \overline{\lambda}x$.
\item If $\lambda_1$ and $\lambda_2$ are distinct eigenvalues of $T$ with corresponding eigenvectors $x_1$ and $x_2$, then $x_1$ and $x_2$ are orthogonal.
\end{enumerate}
\end{theorem}
\begin{theorem}
\hfill\\
Let $T$ be a linear operator on a finite-dimensional complex inner product space $V$. Then $T$ is normal if and only if there exists an orthonormal basis for $V$ consisting of eigenvectors of $T$.
\end{theorem}
\begin{definition}
\hfill\\
Let $T$ be a linear operator on an inner product space $V$. We say that $T$ is \textbf{self-adjoint} (\textbf{Hermitian}) if $T = T^*$. An $n \times n$ real or complex matrix $A$ is \textbf{self-adjoint} (\textbf{Hermitian}) if $A = A^*$.
\end{definition}
\begin{lemma}
\hfill\\
Let $T$ be a self-adjoint operator on a finite-dimensional inner product space $V$. Then
\begin{enumerate}
\item Every eigenvalue of $T$ is real.
\item Suppose that $V$ is a real inner product space. Then the characteristic polynomial of $T$ splits.
\end{enumerate}
\end{lemma}
\begin{theorem}
\hfill\\
Let $T$ be a linear operator on a finite-dimensional real inner product space $V$. Then $T$ is self-adjoint if and only if there exists an orthonormal basis $\beta$ for $V$ consisting of eigenvectors of $T$.
\end{theorem}
\begin{definition}
\hfill\\
An $n \times n$ real matrix $A$ is said to be a \textbf{Gramian} matrix if there exists a real (square) matrix $B$ such that $A = B^tB$.
\end{definition}
\begin{definition}
\hfill\\
A linear operator $T$ on a finite-dimensional inner product space is called \textbf{positive definite [positive semidefinite]} if $T$ is self-adjoint and $\lr{T(x),x} > 0$ [$\lr{T(x),x} \geq 0$] for all $x \neq 0$.
An $n \times n$ matrix $A$ with entries from $\R$ or $\C$ is called \textbf{positive definite [positive semidefinite]} if $L_A$ is positive definite [positive semidefinite].
\end{definition}
chapter-6/orthogonal-projections-and-the-spectral-theorem.tex
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\section{Orthogonal Projections and the Spectral Theorem}
\begin{definition}
\hfill\\
Let $V$ be an inner product space, and let $T: V \to V$ be a projection. We say that $T$ is an \textbf{orthogonal projection} if $\range{T}^\perp = \n{T}$ and $\n{T}^\perp = \range{T}$.
\end{definition}
\begin{definition}
\hfill\\
Let $W$ be a finite-dimensional subspace of an inner product space $V$, and let $T$ be an orthogonal projection on $W$. We call $T$ the \textbf{orthogonal projection of $V$ on $W$}.
\end{definition}
\begin{definition}
A \textbf{trigonometric polynomial of degree $n$} is a function $g \in \mathsf{H}$ of the form
\[g(t) = \sum_{j=-n}^{n}a_jf_j(t) = \sum_{j=-n}^{n}a_je^{ijt},\]
where $a_n$ or $a_{-n}$ is nonzero.
\end{definition}
\begin{theorem}
\hfill\\
Let $V$ be an inner product space, and let $T$ be a linear operator on $V$. Then $T$ is an orthogonal projection if and only if $T$ has an adjoint $T^*$ and $T^2 = T = T^*$.
\end{theorem}
\begin{theorem}[\textbf{The Spectral Theorem}]\label{The Spectral Theorem}
\hfill\\
Suppose that $T$ is a linear operator on a finite-dimensional inner product space $V$ over $\F$ with the distinct eigenvalues $\lambda_1, \lambda_2, \dots, \lambda_k$. Assume that $T$ is normal if $\F = \C$ and that $T$ is self-adjoint if $\F = \R$. For each $i$ ($1 \leq i \leq k$), let $W_i$ be the eigenspace of $T$ corresponding to the eigenvalue $\lambda_i$, and let $T_i$ be the orthogonal projection of $V$ on $W_i$. Then the following statements are true.
\begin{enumerate}
\item $V = W_1 \oplus W_2 \oplus \dots \oplus W_k$.
\item If $W_i'$ denotes the direction sum of the subspaces $W_j$ for $j \neq i$, then $W_i^\perp = W_i'$.
\item $T_iT_j = \delta_{ij}T_i$ for $1 \leq i, j \leq k$.
\item $I = T_1 + T_2 + \dots + T_k$.
\item $T = \lambda_1T_1 + \lambda_2T_2 + \dots + \lambda_kT_k$.
\end{enumerate}
\end{theorem}
\begin{definition}
\hfill\\
In the context of \autoref{The Spectral Theorem}:
\begin{enumerate}
\item The set $\{\lambda_1, \lambda_2, \dots, \lambda_k\}$ of eigenvalues of $T$ is called the \textbf{spectrum} of $T$.
\item The sum $I = T_1 + T_2 + \dots + T_k$ in (4) is called the \textbf{resolution of the identity operator} induced by $T$.
\item The sum $T = \lambda_1T_1 + \lambda_2T_2 + \dots + \lambda_kT_k$ in (5) is called the \textbf{spectral decomposition} of $T$. The spectral decomposition of $T$ is unique up to the order of its eigenvalues.
\end{enumerate}
\end{definition}
\begin{corollary}
\hfill\\
If $\F = \C$, then $T$ is normal if and only if $T^* = g(T)$ for some polynomial $g$.
\end{corollary}
\begin{corollary}
\hfill\\
If $\F = \C$, then $T$ is unitary if and only if $T$ is normal and $|\lambda| = 1$ for every eigenvalue of $T$.
\end{corollary}
\begin{corollary}
\hfill\\
If $\F = \C$ and $T$ is normal, then $T$ is self-adjoint if and only if every eigenvalue of $T$ is real.
\end{corollary}
\begin{corollary}
\hfill\\
Let $T$ be as in \autoref{The Spectral Theorem} with spectral decomposition $T = \lambda_1T_1 + \lambda_2T_2 + \dots + \lambda_kT_k$. Then each $T_j$ is a polynomial in $T$.
\end{corollary}
chapter-6/the-adjoint-of-a-linear-operator.tex
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\section{The Adjoint of a Linear Operator}
\begin{definition}\label{Definition 6.16}
\hfill\\
For a linear operator $T$ on an inner product space $V$, we define a related linear operator on $V$ called the \textbf{adjoint} of $T$, whose matrix representation with respect to any orthonormal basis $\beta$ for $V$ is $[T]_\beta^*$.
\end{definition}
\begin{theorem}
\hfill\\
Let $V$ be a finite-dimensional inner product space over $\F$, and let $\mathsf{g}: V \to \F$ be a linear transformation. Then there exists a unique vector $y \in V$ such that $\mathsf{g}(x) = \lr{x,y}$ for all $x \in V$.
\end{theorem}
\begin{theorem}
\hfill\\
Let $V$ be a finite-dimensional inner product space, and let $T$ be a linear operator on $V$. Then there exists a unique function $T^*: V \to V$ such that $\lr{T(x), y} = \lr{x, T^*(y)}$ for all $x,y \in V$. Furthermore, $T^*$ is linear.
\end{theorem}
\begin{remark}
\hfill\\
\textbf{Important Note:} For the remainder of this chapter we adopt the convention that a reference to the adjoint of a linear operator on an infinite-dimensional inner product space assumes its existence.
\end{remark}
\begin{theorem}
\hfill\\
Let $V$ be a finite-dimensional inner product space, and let $\beta$ be an orthonormal basis for $V$. If $T$ is a linear operator on $V$, then
\[[T^*]_\beta = [T]_\beta^*.\]
\end{theorem}
\begin{corollary}
\hfill\\
Let $A$ be an $n \times n$ matrix. Then $L_{A^*} = (L_A)^*$.
\end{corollary}
\begin{theorem}
\hfill\\
Let $V$ be an inner product space, and let $T$ and $U$ be linear operators on $V$. Then
\begin{enumerate}
\item $(T+U)^* = T^* + U^*$;
\item $(cT)^* = \overline{c}T^*$ for any $c \in \F$;
\item $(TU)^* = U^*T^*$;
\item $T^{**} = T$;
\item $I^* = I$.
\end{enumerate}
\end{theorem}
\begin{corollary}
\hfill\\
Let $A$ and $B$ be $n \times n$ matrices. Then
\begin{enumerate}
\item $(A + B)^* = A^* + B^*$;
\item $(cA)^* = \overline{c}A^*$ for all $c \in \F$;
\item $(AB)^* = B^*A^*$;
\item $A^{**} = A$;
\item $I^* = I$.
\end{enumerate}
\end{corollary}
\begin{lemma}
\hfill\\
Let $A \in M_{m \times n}(\F), x \in F^n$, and $y \in F^m$. Then
\[\lr{Ax,y}_m = \lr{x,A^*y}_n.\]
\end{lemma}
\begin{lemma}
\hfill\\
Let $A \in M_{m \times n}(\F)$. Then $\rank{A^*A} = \rank{A}$.
\end{lemma}
\begin{corollary}
\hfill\\
If $A$ is an $m \times n$ matrix such that $\rank{A} = n$, then $A^*A$ is invertible.
\end{corollary}
\begin{theorem}
\hfill\\
Let $A \in M_{m \times n}(\F)$ and $y \in F^m$. Then there exists $x_0 \in F^n$ such that $(A^*A)x_0 = A^*y$ and $||Ax_0 = y|| \leq ||Ax - y||$ for all $x \in F^n$. Furthermore, if $\rank{A} = n$, then $x_0 = (A^*A)^{-1}A^*y$.
\end{theorem}
\begin{definition}
\hfill\\
A solution $s$ to a system of linear equations $Ax = b$ is called a \textbf{minimal solution} if $||s|| \leq ||u||$ for all other solutions $u$.
\end{definition}
\begin{theorem}
\hfill\\
Let $A \in M_{m \times n}(\F)$ and $b \in F^m$. Suppose that $Ax = b$ is consistent. Then the following statements are true.
\begin{enumerate}
\item There exists exactly one minimal solution $s$ of $Ax = b$, and $x \in \range{L_{A^*}}$.
\item The vector $s$ is the only solution to $Ax = b$ that lies in $\range{L_{A^*}}$; that is, if $u$ satisfies $(AA^*)u = b$, then $s = A^*u$.
\end{enumerate}
\end{theorem}
\begin{definition}
\hfill\\
Let $T: V \to W$ be a linear transformation, where $V$ and $W$ are finite-dimensional inner product spaces with inner products $\lr{\cdot, \cdot}_1$ and $\lr{\cdot, \cdot}_2$, respectively. A function $T^*: W \to V$ is called an \textbf{adjoint} of $T$ if $\lr{T(x),y}_2 = \lr{x,T^*{y}}_1$ for all $x \in V$ and $y \in W$.\\
This is an extension of the definition of the \textit{adjoint} of a linear operator (\autoref{Definition 6.16})
\end{definition}
\begin{definition}
\hfill\\
In physics, \textit{Hooke's law} states that (within certain limits) there is a linear relationship between the length $x$ of a spring and the force $y$ applied to (or exerted by) the spring. That is, $y = cx + d$, where $c$ is called the \textbf{spring constant}.
\end{definition}
chapter-6/the-geometry-of-orthogonal-operators.tex
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\section{The Geometry of Orthogonal Operators}
\begin{definition}
\hfill\\
Let $T$ be a linear operator on a finite-dimensional real inner product space $V$. The operator $T$ is called a \textbf{rotation} if $T$ is the identity on $V$ or if there exists a two-dimensional subspace $W$ of $V$, and orthonormal basis $\beta = \{x_1, x_2\}$ for $W$, and a real number $\theta$ such that
\[T(x_1) = (\cos(\theta))x_1 + (\sin(\theta))x_2,\ \ \ \ T(x_2) = (-\sin(\theta))x_1 + (\cos(\theta))x_2,\]
and $T(y) = y$ for all $y \in W^\perp$. In this context, $T$ is called a \textbf{rotation of $W$ about $W^\perp$}. The subspace $W^\perp$ is called the \textbf{axis of rotation}.
\end{definition}
\begin{definition}
\hfill\\
Let $T$ be a linear operator on a finite-dimensional real inner product space $V$. The operator $T$ is called a \textbf{reflection} if there exists a one-dimensional subspace $W$ of $V$ such that $T(x) = -x$ for all $x \in W$ and $T(y) = y$ for all $y \in W^\perp$. In this context, $T$ is called a \textbf{reflection of $V$ about $W^\perp$}.
\end{definition}
\begin{theorem}
\hfill\\
Let $T$ be an orthogonal operator on a two-dimensional real inner product space $V$. Then $T$ is either a rotation or a reflection. Furthermore, $T$ is a rotation if and only if $\det(T) = 1$, and $T$ is a reflection if and only if $\det(T) = -1$.
\end{theorem}
\begin{corollary}
\hfill\\
Let $V$ be a two-dimensional real inner product space. The composite of a reflection and a rotation on $V$ is a reflection on $V$.
\end{corollary}
\begin{lemma}
\hfill\\
If $T$ is a linear operator on a nonzero finite-dimensional real vector space $V$, then there exists a $T$-invariant subspace $W$ of $V$ such that $1 \leq \ldim{W} \leq 2$.
\end{lemma}
\begin{theorem}\label{Theorem 6.46}
\hfill\\
Let $T$ be an orthogonal operator on a nonzero finite-dimensional real inner product space $V$. Then there exists a collection of pairwise orthogonal $T$-invariant subspaces $\{W_1, W_2, \dots, W_m\}$ of $V$ such that
\begin{enumerate}
\item $1 \leq \ldim(W_i) \leq 2$ for $i = 1, 2, \dots, m$.
\item $V = W_1 \oplus W_2 \oplus \dots \oplus W_m$.
\end{enumerate}
\end{theorem}
\begin{theorem}
\hfill\\
Let $T,V,W_1,\dots,W_m$ be as in \autoref{Theorem 6.46}.
\begin{enumerate}
\item The number of $W_i$'s for which $T_{W_i}$ is a reflection is even or odd according to whether $\det(T) = 1$ or $\det(T) = -1$.
\item It is always possible to decompose $V$ as in \autoref{Theorem 6.46} so that the number of $W_i$'s for which $T_{W_i}$ is a reflection is zero or one according to whether $\det(T) = 1$ or $\det(T) = -1$. Furthermore, if $T_{W_i}$ is a reflection, then $\ldim{W_i} = 1$.
\end{enumerate}
\end{theorem}
\begin{corollary}
\hfill\\
Let $T$ be an orthogonal operator on a finite-dimensional real inner product space $V$. Then there exists a collection $\{T_1, T_2, \dots, T_m\}$ of orthogonal operators on $V$ such that the following statements are true.
\begin{enumerate}
\item For each $i$, $T_i$ is either a reflection or a rotation.
\item For at most one $i$, $T_i$ is a reflection.
\item $T_iT_j = T_jT_i$ for all $i$ and $j$.
\item $T = T_1T_2\dots T_m$.
\item $\det(T) = \displaystyle\begin{cases}
1 & \text{if}\ T_i\ \text{is a rotation for each}\ i \\
-1 & \text{otherwise}
\end{cases}$
\end{enumerate}
\end{corollary}
chapter-6/the-gram-schmidt-orthogonalization-process-and-orthogonal-complements.tex
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\section{The Gram-Schmidt Orthogonalization Process and Orthogonal Complements}
\begin{definition}
\hfill\\
Let $V$ be an inner product space. A subset of $V$ is an \textbf{orthonormal basis} for $V$ if it is an ordered basis that is orthonormal.
\end{definition}
\begin{theorem}\label{Theorem 6.3}
\hfill\\
Let $V$ be an inner product space and $S=\{v_1, v_2, \dots, v_k\}$ be an orthogonal subset of $V$ consisting of nonzero vectors. If $y \in \lspan{S}$, then
\[y = \sum_{i=1}^{k}\frac{\lr{y,v_i}}{||v_i||^2}v_i.\]
\end{theorem}
\begin{corollary}
\hfill\\
If, in addition to the hypotheses of \autoref{Theorem 6.3}, $S$ is orthonormal and $y \in \lspan(S)$, then
\[y = \sum_{i=1}^{k}\lr{y,v_i}v_i.\]
\end{corollary}
\begin{corollary}
\hfill\\
Let $V$ be an inner product space, and let $S$ be an orthogonal subset of $V$ consisting of nonzero vectors. Then $S$ is linearly independent.
\end{corollary}
\begin{theorem}
\hfill\\
Let $V$ be an inner product space and $S = \{w_1, w_2, \dots, w_n\}$ be a linearly independent subset of $V$. Define $S' = \{v_1, v_2, \dots, v_n\}$ where $v_1 = w_1$ and
\[v_k = w_k - \sum_{j=1}^{k-1}\frac{\lr{w_k,v_j}}{||v_j||^2}v_j\ \ \text{for}\ 2 \leq k \leq n.\]
Then $S'$ is an orthogonal set of nonzero vectors such that $\lspan{S'} = \lspan{S}$.
\end{theorem}
\begin{definition}
\hfill\\
The \textbf{Gram-Schmidt process} is a way of making two or more vectors perpendicular to each other. It is a method of constructing an orthonormal basis from a set of vectors in an inner product space. It takes a finite, linearly independent set of vectors $S = \{v_1, v_2, \dots, v_k\}$ for $k \leq n$ and generates an orthogonal set $S' = \{u_1, u_2, \dots, u_k\}$ that spans the same $k$-dimensional subspace of $\R^n$ as $S$.\\
The \textbf{vector projection} of a vector $v$ on a nonzero vector $u$ is defined as
\[\text{proj}_u(v) = \frac{\lr{v,u}}{\lr{u,u}}u,\]
This means that $\text{proj}_u(v)$ is the orthogonal projection of $v$ onto the line spanned by $u$. If $u$ is the zero vector, then $\text{proj}_u(v)$ is defined as the zero vector.\\
Given $k$ vectors $v_1, v_2, \dots, v_k$, the Gram-Schmidt process defines the vectors $u_1, u_2, \dots, u_k$ as follows:
\[\begin{aligned}
& u_1 = v_1, & e_1 = \frac{u_1}{||u_1||} \\
& u_2 = v_2 - \text{proj}_{u_1}(v_2), & e_2 = \frac{u_2}{||u_2||} \\
& u_3 = v_3 - \text{proj}_{u_1}(v_3) - \text{proj}_{u_2}(v_3), & e_3 = \frac{u_3}{||u_3||} \\
& u_4 = v_4 - \text{proj}_{u_1}(v_4) - \text{proj}_{u_2}(v_4) - \text{proj}_{u_3}(v_4), & e_4 = \frac{u_4}{||u_4||} \\
& \vdots & \vdots \\
& u_k = v_k = \sum_{j=1}^{k-1}\text{proj}_{u_j}(v_k), & e_k = \frac{u_k}{||u_k||}.
\end{aligned}\]
The sequence $u_1, u_2, \dots, u_k$ is the required system of orthogonal vectors, and the normalized vectors $e_1, e_2, \dots, e_k$ form an orthonormal set.
\end{definition}
\begin{theorem}
\hfill\\
Let $V$ be a nonzero finite-dimensional inner product space. Then $V$ has an orthonormal basis $\beta$. Furthermore, if $\beta = \{v_1, v_2, \dots, v_n\}$ and $x \in V$, then
\[x = \sum_{i=1}^{n}\lr{x,v_i}v_i.\]
\end{theorem}
\begin{corollary}
\hfill\\
Let $V$ be a finite-dimensional inner product space with an orthonormal basis $\beta = \{v_1, v_2, \dots, v_n\}$. Let $T$ be a linear operator on $V$, and let $A = [T]_\beta$. Then for any $i$ and $j$, $A_{ij} = \lr{T(v_j),v_i}$.
\end{corollary}
\begin{definition}
\hfill\\
Let $\beta$ be an orthonormal subset (possibly infinite) of an inner product space $V$, and let $x \in V$. We define the \textbf{Fourier coefficients} of $x$ relative to $\beta$ to be the scalars $\lr{x,y}$, where $y \in \beta$.
\end{definition}
\begin{definition}
\hfill\\
Let $S$ be a nonempty subset of an inner product space $V$. We define $S^\perp$ (read ``S perp") to be the set of all vectors in $V$ that are orthogonal to every vector in $S$; that is, $S^\perp = '{x \in V : \lr{x,y} = 0,\ \forall y \in S}$. The set $S^\perp$ is called the \textbf{orthogonal complement of $S$}.
\end{definition}
\begin{theorem}\label{Theorem 6.6}
\hfill\\
Let $W$ be a finite-dimensional subspace of an inner product space $V$, and let $y \in V$. Then there exist unique vectors $u \in W$ and $z \in W^\perp$ such that $y = u + z$. Furthermore, if $\{v_1, v_2, \dots, v_k\}$ is an orthonormal basis for $W$, then
\[u = \sum_{i=1}^{k}\lr{y,v_i}v_i.\]
\end{theorem}
\begin{corollary}
\hfill\\
In the notation of \autoref{Theorem 6.6}, the vector $u$ is the unique vector in $W$ that is ``closest" to $y$; that is, for any $x \in W$, $||y - x|| \geq ||y - u||$, and this inequality is an equality if and only if $x = u$.
\end{corollary}
\begin{theorem}
\hfill\\
Suppose that $S = \{v_1, v_2, \dots, v_k\}$ is an orthonormal set in an $n$-dimensional inner product space $V$. Then
\begin{enumerate}
\item $S$ can be extended to an orthonormal basis $\{v_1, v_2, \dots, v_k, v_{k+1}, \dots, v_n\}$ for $V$.
\item If $W = \lspan{S}$, then $S_1 = \{v_{k+1}, v_{k+2}, \dots, v_n\}$ is an orthonormal basis for $W^\perp$ (using the preceding notation).
\item If $W$ is any subspace of $V$, then $\ldim{V} = \ldim{W} + \ldim{W^\perp}$.
\end{enumerate}
\end{theorem}
\begin{definition}[\textbf{Parseval's Identity}]
\hfill\\
Let $V$ be a finite-dimensional inner product space over $\F$, and let $\{v_1, v_2, \dots, v_n\}$ be an orthonormal basis for $V$. Then for any $x,y \in V$,
\[\lr{x,y} = \sum_{i=1}^{n}\lr{x,v_i}\overline{\lr{y,v_i}}.\]
\end{definition}
\begin{definition}[\textbf{Bessel's Inequality}]
Let $V$ be an inner product space, and let $S = \{v_1, v_2, \dots, v_n\}$ be an orthonormal subset of $V$. For any $x \in V$,
\[||x||^2 \geq \sum_{i=1}^{n}|\lr{x,v_i}|^2.\]
\end{definition}
chapter-6/the-singular-value-decomposition-and-the-pseudoinverse.tex
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\section{The Singular Value Decomposition and the Pseudoinverse}
\begin{theorem}[\textbf{Singular Value Theorem for Linear Transformations}]\label{Theorem 6.26}
\hfill\\
Let $V$ and $W$ be finite-dimensional inner product spaces, and let $T: V \to W$ be a linear transformation of rank $r$. Then there exist orthonormal bases $\{v_1, v_2, \dots, v_n\}$ for $V$ and $\{u_1, u_2, \dots, u_m\}$ for $W$ and positive scalars $\sigma_1 \geq \sigma_2 \geq \dots \geq \sigma_r$ such that
\[T(v_i) = \begin{cases}
\sigma_iu_i & \text{if}\ 1 \leq i \leq r \\
0 & \text{if}\ i > r.
\end{cases}\]
Conversely, suppose that the preceding conditions are satisfied. Then for $1 \leq i \leq n$, $v_i$ is an eigenvector of $T^*T$ with corresponding eigenvalue $\sigma_i^2$ if $1 \leq i \leq r$ and $0$ if $i > r$. Therefore the scalars $\sigma_1, \sigma_2, \dots, \sigma_r$ are uniquely determined by $T$.
\end{theorem}
\begin{definition}
\hfill\\
The unique scalars $\sigma_1,\sigma_2,\dots\sigma_r$ in \autoref{Theorem 6.26} are called the \textbf{singular values} of $T$. If $r$ is less than both $m$ and $n$, then the term \textit{singular value} is extended to include $\sigma_{r + 1} = \dots = \sigma_k = 0$, where $k$ is the minimum of $m$ and $n$.
\end{definition}
\begin{definition}
\hfill\\
Let $A$ be an $m \times n$ matrix. We define the \textbf{singular values} of $A$ to be the singular values of the linear transformation $L_A$.
\end{definition}
\begin{theorem}[\textbf{Singular Value Decomposition Theorem for Matrices}]\label{Theorem 6.27}
\hfill\\
Let $A$ be an $m \times n$ matrix of rank $r$ with the positive singular values $\sigma_1 \geq \sigma_2 \geq \dots \geq \sigma_r$, and let $\Sigma$ be the $m \times n$ matrix defined by
\[\Sigma_{ij} = \begin{cases}
\sigma_i & \text{if}\ i = j \leq r \\
0 & \text{otherwise}.
\end{cases}\]
Then there exists an $m \times m$ unitary matrix $U$ and an $n \times n$ unitary matrix $V$ such that
\[A = U\Sigma V^*.\]
\end{theorem}
\begin{definition}
\hfill\\
Let $A$ be an $m \times n$ matrix of rank $r$ with positive singular values $\sigma_1 \geq \sigma_2 \geq \dots \geq \sigma_r$. A factorization $A = U\Sigma V^*$ where $U$ and $V$ are unitary matrices and $\Sigma$ is the $m \times n$ matrix defined as in \autoref{Theorem 6.27} is called a \textbf{singular value decomposition} of $A$.
\end{definition}
\begin{theorem}
\hfill\\
For any square matrix $A$, there exists a unitary matrix $W$ and a positive semidefinite matrix $P$ such that
\[A = WP.\]
Furthermore, if $A$ is invertible, then the representation is unique.
\end{theorem}
\begin{definition}
\hfill\\
The factorization of a square matrix $A$ as $WP$ where $W$ is unitary and $P$ is positive semidefinite is called a \textbf{polar decomposition} of $A$.
\end{definition}
\begin{definition}
\hfill\\
Let $V$ and $W$ be finite-dimensional inner product spaces over the same field, and let $T: V \to W$ be a linear transformation. Let $L: \n{T}^\perp \to \range{T}$ be the linear transformation defined by $L(x) = T(x)$ for all $x \in \n{T}^\perp$. The \textbf{pseudoinverse} (or \textit{Moore-Penrose generalized inverse}) of $T$, denoted by $T^\dagger$, is defined as the unique linear transformation from $W$ to $V$ such that
\[T^\dagger(y) = \begin{cases}
L^{-1}(y) & \text{for}\ y \in \range{T} \\
0 & \text{for}\ y \in \range{T}^\perp.
\end{cases}\]
\end{definition}
\begin{definition}
\hfill\\
Let $A$ be an $m \times n$ matrix. Then there exists a unique $n \times m$ matrix $B$ such that $(L_A)^\dagger: F^m \to F^n$ is equal to the left-multiplication transformation $L_B$. We call $B$ the \textbf{pseudoinverse} of $A$ and denote it by $B = A^\dagger$.
\end{definition}
\begin{theorem}
\hfill\\
Let $A$ be an $m \times n$ matrix of rank $r$ with a singular value decomposition $A = U\Sigma V^*$ and nonzero singular values $\sigma_1 \geq \sigma_2 \geq \dots \geq \sigma_r$. Let $\Sigma^\dagger$ be the $n \times m$ matrix defined by
\[\Sigma_{ij}^\dagger = \begin{cases}
\frac{1}{\sigma_i} & \text{if}\ i = j \leq r \\
0 & \text{otherwise.}
\end{cases}\]
Then $A^\dagger = V\Sigma^\dagger U^*$, and this is a singular value decomposition of $A^\dagger$.
\end{theorem}
\begin{lemma}
\hfill\\
Let $V$ and $W$ be finite-dimensional inner product spaces, and let $T: V \to W$ be linear. Then
\begin{enumerate}
\item $T^\dagger T$ is the orthogonal projection of $V$ on $\n{T}^\dagger$.
\item $TT^\dagger$ is the orthogonal projection of $W$ on $\range{T}$.
\end{enumerate}
\end{lemma}
\begin{theorem}
\hfill\\
Consider the system of linear equations $Ax = b$, where $A$ is an $m \times n$ matrix and $b \in F^m$. If $z = A^\dagger b$, then $z$ has the following properties.
\begin{enumerate}
\item If $Ax = b$ is consistent, then $z$ is the unique solution to the system having minimum norm. That is, $z$ is a solution to the system, and if $y$ is any solution to the system, then $||z|| \leq ||y||$ with equality if and only if $z = y$.
\item If $Ax = b$ is inconsistent, then $z$ is the unique best approximation to a solution having minimum norm. That is, $||Az - b|| \leq ||Ay - b||$ for any $y \in F^n$, with equality if and only if $Az = Ay$. Furthermore, if $Az = Ay$, then $||z|| \leq ||y||$ with equality if and only if $z = y$.
\end{enumerate}
\end{theorem}
\begin{lemma}[\textbf{Penrose Conditions}]
\hfill\\
Let $V$ and $W$ be finite-dimensional inner product spaces, and let $T: V \to W$ be linear. Then the following conditions hold:
\begin{enumerate}
\item $TT^\dagger T = T$.
\item $T^\dagger TT^\dagger = T^\dagger$.
\item Both $T^\dagger T$ and $TT^\dagger$ are self-adjoint.
\end{enumerate}
These conditions characterize the pseudoinverse of a linear transformation.
\end{lemma}
chapter-6/unitary-and-orthogonal-operators-and-their-matrices.tex
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\section{Unitary and Orthogonal Operators and Their Matrices}
\begin{definition}
\hfill\\
Let $T$ be a linear operator on a finite-dimensional inner product space $V$ (over $\F$). If $||T(x)|| = ||x||$ for all $x \in V$, we call $T$ a \textbf{unitary operator} if $\F = \C$ and an \textbf{orthogonal operator} if $\F = \R$.\\
It should be noted that, in the infinite-dimensional case, an operator satisfying the preceding norm requirement is generally called an \textbf{isometry}. If, in addition, the operator is onto (the condition guarantees one-to-one), then the operator is called a \textbf{unitary} or \textbf{orthogonal operator},
\end{definition}
\begin{theorem}
\hfill\\
Let $T$ be a linear operator on a finite-dimensional inner product space $V$. Then the following statements are equivalent.
\begin{enumerate}
\item $TT^* = T^*T = I$.
\item $\lr{T(x), T(y)} = \lr{x,y}$ for all $x,y \in V$.
\item If $\beta$ is an orthonormal basis for $V$, then $T(\beta)$ is an orthonormal basis for $V$.
\item There exists an orthonormal basis $\beta$ for $V$ such that $T(\beta)$ is an orthonormal basis for $V$.
\item $||T(x)|| = ||x||$ for all $x \in V$.
\end{enumerate}
\end{theorem}
\begin{lemma}
\hfill\\
Let $U$ be a self-adjoint operator on a finite-dimensional inner product space $V$. If $\lr{x,U(x)} = 0$ for all $x \in V$, then $U = T_0$.
\end{lemma}
\begin{corollary}
\hfill\\
Let $T$ be a linear operator on a finite-dimensional real inner product space $V$. Then $V$ has an orthonormal basis of eigenvectors of $T$ with corresponding eigenvalues of absolute value $1$ if and only if $T$ is both self-adjoint and orthogonal.
\end{corollary}
\begin{corollary}
\hfill\\
Let $T$ be a linear operator on a finite-dimensional complex inner product space $V$. Then $V$ has an orthonormal basis of eigenvectors of $T$ with corresponding eigenvalues of absolute value $1$ if and only if $T$ is unitary.
\end{corollary}
\begin{definition}
\hfill\\
Let $L$ be a one-dimensional subspace of $\R^2$. We may view $L$ as a line in the plane through the origin. A linear operator $T$ on $\R^2$ is called a \textbf{reflection of $\R^2$ about $L$} if $T(x) = x$ for all $x \in L$ and $T(x) = -x$ for all $x \in L^\perp$.
\end{definition}
\begin{definition}
\hfill\\
A square matrix $A$ is called an \textbf{orthogonal matrix} if $A^tA = AA^t = I$ and \textbf{unitary} if $A^*A = AA^* = I$.\\
Since for a real matrix $A$ we have $A^* = A^t$, a real unitary matrix is also orthogonal. In this case, we call $A$ \textbf{orthogonal} rather than unitary.
\end{definition}
\begin{definition}
\hfill\\
We know that, for a complex normal [real symmetric] matrix $A$, there exists an orthonormal basis $\beta$ for $F^n$ consisting of eigenvectors of $A$. Hence $A$ is similar to a diagonal matrix $D$. By \autoref{Corollary 2.8}, the matrix $Q$ whose columns are the vectors in $\beta$ is such that $D = Q^{-1}AQ$. But since the columns of $Q$ are an orthonormal basis for $F^n$, it follows that $Q$ is unitary [orthogonal]. In this case, we say that $A$ is \textbf{unitarily equivalent} [\textbf{orthogonally equivalent}] to $D$. It is easily seen that this relation is an equivalence relation on $M_{n \times n}(\C)$ [$M_{n \times n}(\R)$]. More generally, \textit{$A$ and $B$ are unitarily equivalent [orthogonally equivalent]} if and only if there exists a unitary [orthogonal] matrix $P$ such that $A = P^*BP$.
\end{definition}
\begin{theorem}
\hfill\\
Let $A$ be a complex $n \times n$ matrix. Then $A$ is normal if and only if $A$ is unitarily equivalent to a diagonal matrix.
\end{theorem}
\begin{theorem}
\hfill\\
Let $A$ be a real $n \times n$ matrix. Then $A$ is symmetric if and only if $A$ is orthogonally equivalent to a real diagonal matrix.
\end{theorem}
\begin{theorem}[\textbf{Schur}]
\hfill\\
Let $A \in M_{n \times n}(\F)$ be a matrix whose characteristic polynomial splits over $\F$.
\begin{enumerate}
\item If $\F = \C$, then $A$ is unitarily equivalent to a complex upper triangular matrix.
\item If $\F = \R$, then $A$ is orthogonally equivalent to a real upper triangular matrix.
\end{enumerate}
\end{theorem}
\begin{definition}
\hfill\\
Let $V$ be a real inner product space. A function $f: V \to V$ is called a \textbf{rigid motion} if
\[||f(x) - f(y)|| = ||x - y||\]
for all $x,y \in V$.
\end{definition}
\begin{definition}
\hfill\\
Let $V$ be a real inner product space. A function $g: V \to V$ is called a \textbf{translation} if there exists a vector $v_0 \in V$ such that $g(x) = x + v_0$ for all $x \in V$. We say that $g$ is the \textit{translation by $v_0$}.
\end{definition}
\begin{theorem}
\hfill\\
Let $f: V \to V$ be a rigid motion on a finite-dimensional real inner product space $V$. Then there exists a unique orthogonal operator $T$ on $V$ and a unique translation $g$ on $V$ such that $f = g \circ T$.
\end{theorem}
\begin{theorem}
\hfill\\
Let $T$ be an orthogonal operator on $\R^2$, and let $A = [T]_\beta$ where $\beta$ is the standard ordered basis for $\R^2$. Then exactly one of the following conditions is satisfied:
\begin{enumerate}
\item $T$ is a rotation, and $\det(A) = 1$.
\item $T$ is a reflection about a line through the origin, and $\det(A) = -1$.
\end{enumerate}
\end{theorem}
\begin{corollary}
\hfill\\
Any rigid motion on $\R^2$ is either a rotation followed by a translation or a reflection about a line through the origin followed by a translation.
\end{corollary}
\begin{definition}
Consider the quadratic equation
\begin{equation}\label{eq:quad}
ax^2 + 2bxy + cy^2 +dx + ey + f = 0.
\end{equation}
The expression
\[ax^2 + 2bxy + cy^2\]
is called the \textbf{associated quadratic form} of \eqref{eq:quad}
\end{definition}
\begin{definition}
\hfill\\
Let $V$ be a finite-dimensional inner product space. A linear operator $U$ on $V$ is called a \textbf{partial isometry} if there exists a subspace $W$ of $V$ such that $||U(x)|| = ||x||$ for all $x \in W$ and $U(x) = 0$ for all $x \in W^\perp$.
\end{definition}
\begin{definition}
Let $V$ be a finite-dimensional complex [real] inner product space, and let $u$ be a unit vector in $V$. Define the \textbf{Householder} operator $\mathsf{H}_u: V \to V$ by $\mathsf{H}_u(x) = x-2\lr{x,u}u$ for all $x \in V$.
\end{definition}