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Wrote out chapters 2-4

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Alex Clarke
2024-02-20 16:06:34 -07:00
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chapter-4/a-characterization-of-the-determinant.tex
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\section{A Characterization of the Determinant}
\begin{definition}
\hfill\\
A function $\delta: M_{n \times n}(\F) \to \F$ is called an \textbf{\textit{n}-linear function} if it is a linear function of each row of an $n \times n$ matrix when the remaining $n-1$ rows are held fixed, that is, $\delta$ is $n$-linear if, for every $r = 1, 2, \dots, n$, we have
\[\delta\begin{pmatrix}
a_1 \\ \vdots \\ a_{r-1} \\ u+kv \\ a_{r + 1} \\ \vdots \\ a_n
\end{pmatrix} = \delta\begin{pmatrix}
a_1 \\ \vdots \\ a_{r-1} \\ u \\ a_{r + 1} \\ \vdots \\ a_n
\end{pmatrix} + k\delta\begin{pmatrix}
a_1 \\ \vdots \\ a_{r-1} \\ v \\ a_{r+1} \\ \vdots \\ a_n
\end{pmatrix}\]
whenever $k$ is a scalar and $u,v$ and each $a_i$ are vectors in $\F^n$.
\end{definition}
\begin{definition}
\hfill\\
An $n$-linear function $\delta: M_{n \times n}(\F) \to \F$ is called \textbf{alternating} if, for each $A \ in M_{n \times n}(\F)$, we have $\delta(A) = 0$ whenever two adjacent rows of $A$ are identical.
\end{definition}
\begin{theorem}
\hfill\\
Let $\delta: M_{n \times n}(\F) \to \F$ be an alternating $n$-linear function.
\begin{enumerate}
\item If $A \in M_{n \times n}(\F)$ and $B$ is a matrix obtained from $A$ by interchanging any two rows of $A$, then $\delta(B) = -\delta(A)$.
\item If $A \in M_{n \times n}(\F)$ has two identical rows, then $\delta(A) = 0$.
\end{enumerate}
\end{theorem}
\begin{corollary}
\hfill\\
Let $\delta: M_{n \times n}(\F) \to \F$ be an alternating $n$-linear function. If $B$ is a matrix obtained from $A \in M_{n \times n}(\F)$ by adding a multiple of some row of $A$ to another row, then $\delta(B) = \delta(A)$.
\end{corollary}
\begin{corollary}
\hfill\\
Let $\delta: M_{n \times n}(\F) \to \F$ be an alternating $n$-linear function. if $M \in M_{n \times n}(\F)$ has rank less than $n$, then $\delta(M) = 0$.
\end{corollary}
\begin{corollary}
\hfill\\
Let $\delta: M_{n \times n}(\F) \to \F$ be an alternating $n$-linear function, and let $E_1, E_2$ and $E_3$ in $M_{n \times n}(\F)$ be elementary matrices of types 1, 2, and 3, respectively. Suppose that $E_2$ is obtained by multiplying some row of $I$ by the nonzero scalar $k$. Then $\delta(E_1) = -\delta(I)$, $\delta(E_2) = k \cdot \delta(I)$, and $\delta(E_3) = \delta(I)$.
\end{corollary}
\begin{theorem}
\hfill\\
Let $\delta: M_{n \times n}(\F) \to \F$ be an alternating $n$-linear function such that $\delta(I) = 1$. For any $A,B \in M_{n \times n}(\F)$, we have $\delta(AB) = \delta(A) \cdot \delta(B)$.
\end{theorem}
\begin{theorem}
\hfill\\
If $\delta: M_{n \times n}(\F) \to \F$ is an alternating $n$-linear function such that $\delta(I) = 1$, then $\delta(A) = \det(A)$ for every $A \in M_{n \times n}(\F)$.
\end{theorem}
chapter-4/determinants-of-order-2.tex
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\section{Determinants of Order 2}
\begin{definition}
\hfill\\
If
\[A = \begin{pmatrix}
a & b \\
c & d
\end{pmatrix}\]
is a $2 \times 2$ matrix with entries from a field $\F$, then we define the \textbf{determinant} of $A$, denoted $\det(A)$ or $|A|$, to be the scalar $ad-bc$.
\end{definition}
\begin{theorem}
\hfill\\
The function $\det: M_{2 \times 2}(\F) \to \F$ is a linear function of each row of a $2 \times 2$ matrix when the other row is held fixed. That is, if $u$, $v$ and $w$ are in $\F^2$ and $k$ is a scalar, then
\[\det \begin{pmatrix}
u + kv \\
w
\end{pmatrix} = \det\begin{pmatrix}
u \\ w
\end{pmatrix} + k\det\begin{pmatrix}
v \\ w
\end{pmatrix}\]
and
\[\det\begin{pmatrix}
w \\ u + kv
\end{pmatrix} = \det\begin{pmatrix}
w \\ u
\end{pmatrix} + k \det \begin{pmatrix}
w \\ v
\end{pmatrix}.\]
\end{theorem}
\begin{theorem}\label{Theorem 4.2}
\hfill\\
Let $A \in M_{2 \times 2}(\F)$. Then the determinant of $A$ is nonzero if and only if $A$ is invertible. Moreover, if $A$ is invertible, then
\[A^{-1} = \frac{1}{\det(A)}\begin{pmatrix}
A_{22} & -A_{12} \\
-A_{21} & A_{11}
\end{pmatrix}.\]
\end{theorem}
\begin{definition}
\hfill\\
By the \textbf{angle} between two vectors in $\R^2$, we mean the angle with measure $\theta$ ($0 \leq \theta < \pi$) that is formed by the vectors having the same magnitude and direction as the given vectors by emanating from the origin.
\end{definition}
\begin{definition}
\hfill\\
If $\beta = \{u,v\}$ is an ordered basis for $\R^2$, we define the \textbf{orientation} of $\beta$ to be the real number
\[O\begin{pmatrix}
u \\ v
\end{pmatrix} = \frac{\det\begin{pmatrix}
u \\ v
\end{pmatrix}}{\abs{\det\begin{pmatrix}
u \\ v
\end{pmatrix}}}\]
(The denominator of this fraction is nonzero by \autoref{Theorem 4.2}).
\end{definition}
\begin{definition}
\hfill\\
A coordinate system $\{u, v\}$ is called \textbf{right-handed} if $u$ can be rotated in a counterclockwise direction through an angle $\theta$ ($0 < \theta < \pi$) to coincide with $v$. Otherwise, $\{u ,v\}$ is called a \textbf{left-handed} system.
\end{definition}
\begin{definition}
\hfill\\
Any ordered set $\{u, v\}$ in $\R^2$ determines a parallelogram in the following manner. Regarding $u$ and $v$ as arrows emanating from the origin of $\R^2$, we call the parallelogram having $u$ and $v$ as adjacent sides the \textbf{parallelogram determined by $u$ and $v$}.
\end{definition}
chapter-4/determinants-of-order-n.tex
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\section{Determinants of Order $n$}
\section{Determinants of Order \textit{n}}
\begin{notation}
\hfill\\
Given $A \in M_{n \times n}(\F)$, for $n \geq 2$, denote the $(n-1) \times (n - 1)$ matrix obtained from $A$ by deleting row $i$ and column $j$ by $\tilde{A}_{ij}$. Thus for
\[A = \begin{pmatrix}
1 & 2 & 3 \\
4 & 5 & 6 \\
7 & 8 & 9
\end{pmatrix} \in M_{3 \times 3}(\R)\]
we have
\[\tilde{A}_{11} = \begin{pmatrix}
5 & 6 \\
8 & 9
\end{pmatrix},\ \ \ \ \
\tilde{A}_{13}=\begin{pmatrix}
4 & 5 \\
7 & 8
\end{pmatrix},\ \ \ \ \
\tilde{A}_{32} = \begin{pmatrix}
1 & 3 \\
4 & 6
\end{pmatrix}\]
and for
\[B = \begin{pmatrix}
1 & -1 & 2 & -1 \\
-3 & 4 & 1 & -1 \\
2 & -5 & -3 & 8 \\
-2 & 6 & -4 & 1
\end{pmatrix}\]
we have
\[\tilde{B}_{23} = \begin{pmatrix}
1 & -1 & -1 \\
2 & -5 & 8 \\
-2 & 6 & 1
\end{pmatrix}\ \ \ \ \ \text{and}\ \ \ \ \ \tilde{B}_{42}=\begin{pmatrix}
1 & 2 & -1 \\
-3 & 1 & -1 \\
2 & -3 & 8
\end{pmatrix}\]
\end{notation}
\begin{definition}
\hfill\\
Let $A \in M_{n \times n}(\F)$. If $n =1$, so that $A = (A_{11})$, we define $\det(A) = A_{11}$. For $n \geq 2$, we define $\det(A)$ recursively as
\[\det(A) = \sum_{j=1}^{n}(-1)^{1+j}A_{1j}\cdot\det(\tilde{A}_{1j}).\]
The scalar $\det(A)$ is called the \textbf{determinant} of $A$ and is also denoted by $|A|$. The scalar
\[(-1)^{i+j}\det(\tilde{A}_{ij})\]
is called the \textbf{cofactor} of the entry of $A$ in row $i$, column $j$.
\end{definition}
\begin{definition}
\hfill\\
Letting
\[c_{ij} = (-1)^{i+j}\det(\tilde{A}_{ij})\]
denote the cofactor of the row $i$, column $j$ entry of $A$, we can express the formula for the determinant of $A$ as
\[\det(A) = A_{11}c_{11} + A_{12}c_{12}+\dots+A_{1n}c_{1n}.\]
Thus the determinant of $A$ equals the sum of the products of each entry in row $1$ of $A$ multiplied by its cofactor. This formula is called \textbf{cofactor expansion along the first row} of $A$.
\end{definition}
\begin{theorem}
\hfill\\
the determinant of an $n \times n$ matrix is a linear function of each row when the remaining rows are held fixed. That is, for $1 \leq r \leq n$, we have
\[\det\begin{pmatrix}
a_1 \\ \vdots \\ a_{r-1} \\ u+kv \\ a_{r+1} \\ \vdots \\ a_n
\end{pmatrix}=\det\begin{pmatrix}
a_1 \\ \vdots \\ a_{r-1} \\ u \\ a_{r+1} \\ \vdots \\ a_n
\end{pmatrix} + k\det\begin{pmatrix}
a_1 \\ \vdots \\ a_{r-1} \\ v \\ a_{r+1} \\ \vdots \\ a_n
\end{pmatrix}\]
wherever $k$ is a scalar and $u, v$ and each $a_i$ are row vectors in $\F^n$.
\end{theorem}
\begin{corollary}
\hfill\\
If $A \in M_{n \times n}(\F)$ has a row consisting entirely of zeros, then $\det(A) = 0$.
\end{corollary}
\begin{lemma}
\hfill\\
Let $B \in M_{n \times n}(\F)$, where $n \geq 2$. If row $i$ of $B$ equals $e_k$ for some $k$ ($1 \leq k \leq n$), then $\det(B) = (-1)^{i+k}\det(\tilde{B}_{ik})$.
\end{lemma}
\begin{theorem}
\hfill\\
The determinant of a square matrix can be evaluated by cofactor expansion along any row. That is, if $A \in M_{n \times n}(\F)$, then for any integer $i$ ($1 \leq i \leq n$),
\[\det(A) = \sum_{j=1}^{n}(-1)^{i+j}A_{ij}\cdot\det(\tilde{A}_{ij}).\]
\end{theorem}
\begin{corollary}
\hfill\\
If $A \in M_{n \times n}(\F)$ has two identical rows, then $\det(A) = 0$.
\end{corollary}
\begin{theorem}
\hfill\\
If $A \in M_{n \times n}(\F)$ and $B$ is a matrix obtained from $A$ by interchanging any two rows of $A$, then $\det(B) = -\det(A)$.
\end{theorem}
\begin{theorem}
\hfill\\
Let $A \in M_{n \times n}(\F)$, and let $B$ be a matrix obtained by adding a multiple of one row of $A$ to another row of $A$. Then $\det(B) = \det(A)$.
\end{theorem}
\begin{corollary}
\hfill\\
If $A \in M_{n \times n}(\F)$ has rank less than $n$, then $\det(A) = 0$.
\end{corollary}
\begin{remark}\label{Remark 4.1}
\hfill\\
The following rules summarize the effect of an elementary row operation on the determinant of a matrix $A \ in M_{n \times n}(\F)$.
\begin{enumerate}
\item If $B$ is a matrix obtained by interchanging any two rows of $A$, then $\det(B) = -\det(A)$.
\item If $B$ is a matrix obtained by multiplying a row of $A$ by a nonzero scalar $k$, then $\det(B) = k\det(A)$.
\item If $B$ is a matrix obtained by adding a multiple of one row of $A$ to another row of $A$, then $\det(B) = \det(A)$.
\end{enumerate}
\end{remark}
\begin{lemma}
\hfill\\
The determinant of an upper triangular matrix is the product of its diagonal entries.
\end{lemma}
chapter-4/properties-of-determinants.tex
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\section{Properties of Determinants}
\begin{remark}
\hfill\\
Because the determinant of the $n \times n$ matrix is $1$, we can interpret \autoref{Remark 4.1} as the following facts about the determinants of elementary matrices.
\begin{enumerate}
\item If $E$ is an elementary matrix obtained by interchanging any two rows of $I$, then $\det(E) = -1$.
\item If $E$ is an elementary matrix obtained by multiplying some row of $I$ by the nonzero scalar $k$, then $\det(E) = k$.
\item If $E$ is an elementary matrix obtained by adding a multiple of some row of $I$ to another row, then $\det(E) = 1$.
\end{enumerate}
\end{remark}
\begin{theorem}
\hfill\\
For any $A, B \in M_{n \times n}(\F)$, $\det(AB) = \det(A) \cdot \det(B)$.
\end{theorem}
\begin{corollary}
\hfill\\
A matrix $A \in M_{n \times n}(\F)$ is invertible if and only if $\det(A) \neq 0$. Furthermore, if $A$ is invertible, then $\det(A^{-1}) = \displaystyle\frac{1}{\det(A)}$.
\end{corollary}
\begin{theorem}
\hfill\\
For any $A \in M_{n \times n}(\F)$, $\det(A^t)=\det(A)$.
\end{theorem}
\begin{theorem}[\textbf{Cramer's Rule}]
\hfill\\
Let $Ax = b$ be the matrix form of a system of $n$ linear equations in $n$ unknowns, where $x = (x_1, x_2, \dots, x_n)^t$. If $\det(A) \neq 0$, then this system has a unique solution, and for each $k$ ($k = 1, 2, \dots, n$),
\[x_k = \frac{\det(M_k)}{\det(A)},\]
where $M_k$ is the $n \times n$ matrix obtained from $A$ by replacing column $k$ of $A$ by $b$.
\end{theorem}
\begin{definition}
\hfill\\
It is possible to interpret the determinant of a matrix $A \in M_{n \times n}(\R)$ geometrically. If the rows of $A$ are $a_1, a_2, \dots, a_n$, respectively, then $|\det(A)|$ is the \textbf{\textit{n}-dimensional volume} (the generalization of are in $\R^2$ and volume in $\R^3$) of the parallelepiped having the vectors $a_1, a_2, \dots, a_n$ as adjacent sides.
\end{definition}
\begin{definition}
\hfill\\
A matrix $M \in M_{n \times n}(\C)$ is called \textbf{nilpotent} if, for some positive integer $k$, $M^k = O$, where $O$ is the $n \times n$ zero matrix.
\end{definition}
\begin{definition}
\hfill\\
A matrix $M \in M_{n \times n}(\C)$ is called \textbf{skew-symmetric} if $M^t = -M$.
\end{definition}
\begin{definition}
\hfill\\
A matrix $Q \in M_{n \times n}(\R)$ is called \textbf{orthogonal} if $QQ^t = I$.
\end{definition}
\begin{definition}
\hfill\\
A matrix $Q \in M_{n \times n}(\C)$ is called \textbf{unitary} if $QQ^* = I$, where $Q^* = \overline{Q^t}$.
\end{definition}
\begin{definition}
\hfill\\
A matrix $A \in M_{n \times n}(\F)$ is called \textbf{lower triangular} if $A_{ij}=0$ for $1 \leq i < j \leq n$.
\end{definition}
\begin{definition}
\hfill\\
A matrix of the form
\[\begin{pmatrix}
1 & c_0 & c_0^2 & \dots & c_0^n \\
1 & c_1 & c_1^2 & \dots & c_1^n \\
\vdots & \vdots & \vdots & &\vdots \\
1 & c_n & c_n^2 & \dots & c_n^n
\end{pmatrix}\]
is called a \textbf{Vandermonde matrix}.
\end{definition}
\begin{definition}
\hfill\\
Let $A \in M_{n \times n}(\F)$ be nonzero. For any $m$ ($1 \leq m \leq n$), and $m \times m$ \textbf{submatrix} is obtained by deleting any $n - m$ rows and any $n - m$ columns of $A$.
\end{definition}
\begin{definition}
\hfill\\
The \textbf{classical adjoint} of a square matrix $A$ is the transpose of the matrix whose $ij$-entry is the $ij$-cofactor of $A$.
\end{definition}
\begin{definition}
\hfill\\
Let $y_1, y_2, \dots, y_n$ be linearly independent function in $\C^\infty$. For each $y \in \C^\infty$, define $T(y) \in \C^\infty$ by
\[[T(y)](t) = \det\begin{pmatrix}
y(t) & y_1(t) & y_2(t) & \dots & y_n(t) \\
y'(t) & y'_1(t) & y'_2(t) & \dots & y'_n(t) \\
\vdots & \vdots & \vdots & &\vdots \\
y^{(n)}(t) & y_1^{(n)}(t) & y_2^{(n)}(t) & \dots & y_n^{(n)}(t)
\end{pmatrix}\]
The preceding determinant is called the \textbf{Wronskian} of $y, y_1, \dots, y_n$.
\end{definition}
chapter-4/summary-important-facts-about-determinants.tex
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\section{Summary -- Important Facts about Determinants}
\begin{definition}
\hfill\\
The \textbf{determinant} of an $n \times n$ matrix $A$ having entries from a field $\F$ is a scalar in $\F$, denoted by $\det(A)$ or $|A|$, and can be computed in the following manner:
\begin{enumerate}
\item If $A$ is $1 \times 1$, then $\det(A) = A_{11}$, the single entry of $A$.
\item If $A$ is $2 \times 2$, then $\det(A) = A_{11}A_{22} - A_{12}A_{21}$.
\item If $A$ is $n \times n$ for $n > 2$, then
\[\det(A) = \sum_{j=1}^{n}(-1)^{i+j}A_{ij}\cdot\det(\tilde{A}_{ij})\]
(if the determinant is evaluated by the entries of row $i$ of $A$) or
\[\det(A) = \sum_{i=1}^{n}(-1)^{i+j}A_{ij}\cdot\det(\tilde{A}_{ij})\]
(if the determinant is evaluated by the entries of column $j$ of $A$), where $\tilde{A}_{ij}$ is the $(n-1) \times (n-1)$ matrix obtained by deleting row $i$ and column $j$ from $A$.
\end{enumerate}
In the formulas above, the scalar $(-1)^{i+j}\det(\tilde{A}_{ij})$ is called the \textbf{cofactor} of the row $i$ column $j$ of $A$.
\end{definition}
\begin{definition}[\textbf{Properties of the Determinant}]
\begin{enumerate}
\item[]
\item If $B$ is a matrix obtained by interchanging any two rows or interchanging any two columns of an $n \times n$ matrix $A$, then $\det(B) = -\det(A)$.
\item If $B$ is a matrix obtained by multiplying each entry of some row or column of an $n \times n$ matrix $A$ by a scalar $k$, then $\det(B) = k\cdot\det(A)$.
\item If $B$ is a matrix obtained from an $n \times n$ matrix $A$ by adding a multiple of row $i$ to row $j$ or a multiple of column $i$ to column $j$ for $i \neq j$, then $\det(B) = \det(A)$.
\end{enumerate}
\end{definition}