89 lines
8.7 KiB
TeX
89 lines
8.7 KiB
TeX
\chapter{List of Symbols}
|
|
\begin{align*}
|
|
& A_{ij} & \text{the $ij$-th entry of the matrix $A$} \\
|
|
& A^{-1} & \text{the inverse of the matrix $A$} \\
|
|
& A^\dag & \text{the pseudoinverse of the matrix $A$} \\
|
|
& A^* & \text{the adjoint of the matrix $A$} \\
|
|
& \tilde{A}_{ij} & \text{the matrix $A$ with row $i$ and column $j$ deleted} \\
|
|
& A^t & \text{the transpose of the matrix $A$} \\
|
|
& (A|B) & \text{the matrix $A$ augmented by the matrix $B$} \\
|
|
& B_1 \bigoplus \dots \bigoplus B_k & \text{the direct sum of matrices $B_1$ through $B_k$} \\
|
|
& \mathcal{B}(V) & \text{the set of bilinear forms on $V$} \\
|
|
& \beta^* & \text{the dual basis of $\beta$} \\
|
|
& \beta_x & \text{the $T$-cyclic basis generated by $x$} \\
|
|
& \C & \text{the field of complex numbers} \\
|
|
& \C_i & \text{the $i$th Gerschgorin disk} \\
|
|
& \cond{A} & \text{the condition number of the matrix $A$} \\
|
|
& C^n(\R) & \text{set of functions $f$ on $\R$ with $f^{(n)}$ continuous} \\
|
|
& C^\infty & \text{set of functions with derivatives of every order} \\
|
|
& C(\R) & \text{the vector space of continuous functions on $\R$} \\
|
|
& C([0,1]) & \text{the vector space of continuous functions on $[0,1]$} \\
|
|
& C_x & \text{the $T$-cyclic subspaces generated by $x$} \\
|
|
& D & \text{the derivative operator on $C^\infty$} \\
|
|
& \ldet{A} & \text{the determinant of the matrix $A$} \\
|
|
& \delta_{ij} & \text{the Kronecker delta} \\
|
|
& \ldim{V} & \text{the dimension of $V$} \\
|
|
& e^A & \lim_{m \to \infty} \left(I + A + \frac{A^2}{2!} + \dots + \frac{A^m}{m!}\right) \\
|
|
& e_i & \text{the $i$th standard vector of $\F^n$} \\
|
|
\end{align*}
|
|
\begin{align*}
|
|
& E_\lambda & \text{the eigenspace of $T$ corresponding to $\lambda$} \\
|
|
& \F & \text{a field} \\
|
|
& f(A) & \text{the polynomial $f(x)$ evaluated at the matrix $A$} \\
|
|
& F^n & \text{the set of $n$-tuples with entries in a field $\F$} \\
|
|
& f(T) & \text{the polynomial $f(x)$ evaluated at the operator $T$} \\
|
|
& \mathcal{F}(S,\F) & \text{the set of functions from $S$ to a field $\F$} \\
|
|
& H & \text{space of continuous complex functions on $[0, 2\pi]$} \\
|
|
& I_n \text{ or } I & \text{the $n \times n$ identity matrix} \\
|
|
& \Id_V \text{ or } \Id & \text{the identity operator on $V$} \\
|
|
& K_\lambda & \text{generalized eigenspace of $T$ corresponding to $\lambda$} \\
|
|
& K_\phi & \{x\ |\ (\phi(T))^p(x) = 0 \text{, for some positive integer $p$}\} \\
|
|
& L_A & \text{left-multiplication transformation by matrix $A$} \\
|
|
& \lim_{m \to \infty}A_m & \text{the limit of a sequence of matrices} \\
|
|
& \linear{V} & \text{the space of linear transformations from $V$ to $V$} \\
|
|
& \linear{V, W} & \text{the space of linear transformations from $V$ to $W$} \\
|
|
& M_{m \times n}(\F) & \text{the set of $m \times n$ matrices with entries in $\F$} \\
|
|
& v(A) & \text{the column sum of the matrix $A$} \\
|
|
& v_j(A) & \text{the $j$th column sum of the matrix $A$} \\
|
|
& N(T) & \text{the null space of $T$} \\
|
|
& \nullity{T} & \text{the dimension of the null space of $T$} \\
|
|
& O & \text{the zero matrix} \\
|
|
& \per{M} & \text{the permanent of the $2 \times 2$ matrix $M$} \\
|
|
& P(\F) & \text{the space of polynomials with coefficients in $\F$} \\
|
|
& P_n(\F) & \text{the polynomials in $P(\F)$ of degree at most $n$} \\
|
|
& \phi_\beta & \text{the standard representation with respect to basis $\beta$} \\
|
|
& \R & \text{the field of real numbers} \\
|
|
& \rank{A} & \text{the rank of the matrix $A$} \\
|
|
& \rank{T} & \text{the rank of the linear transformation $T$} \\
|
|
& \rho(A) & \text{the row sum of the matrix $A$} \\
|
|
& \rho_i(A) & \text{the $i$th row sum of the matrix $A$} \\
|
|
& R(T) & \text{the range of the linear transformation $T$} \\
|
|
& S_1 + S_2 & \text{the sum of sets $S_1$ and $S_2$} \\
|
|
& \lspan{S} & \text{the span of the set $S$} \\
|
|
& S^\perp & \text{the orthogonal complement of the set $S$} \\
|
|
& [T]_\beta & \text{the matrix representation of $T$ in basis $\beta$} \\
|
|
& [T]_\beta^\gamma & \text{the matrix representation of $T$ in bases $\beta$ and $\gamma$} \\
|
|
& T^{-1} & \text{the inverse of the linear transformation $T$} \\
|
|
\end{align*}
|
|
\begin{align*}
|
|
& T^\dag & \text{the pseudoinverse of the linear transformation $T$} \\
|
|
& T^* & \text{the adjoint of the linear operator $T$} \\
|
|
& T_0 & \text{the zero transformation} \\
|
|
& T^t & \text{the transpose of the linear transformation $T$} \\
|
|
& T_\theta & \text{the rotation transformation by $\theta$} \\
|
|
& T_W & \text{the restriction of $T$ to a subspace $W$} \\
|
|
& \ltr{A} & \text{the trace of the matrix $A$} \\
|
|
& V^* & \text{the dual space of the vector space $V$} \\
|
|
& V/W & \text{the quotient space of $V$ modulo $W$} \\
|
|
& W_1 + \dots + W_k & \text{the sum of subspaces $W_1$ through $W_k$} \\
|
|
& \sum_{i=1}^k W_i & \text{the sum of subspaces $W_i$ through $W_k$} \\
|
|
& W_1 \bigoplus W_2 & \text{the direct sum of subspaces $W_1$ and $W_2$} \\
|
|
& W_1 \bigoplus \dots \bigoplus W_k & \text{the direct sum of subspaces $W_1$ through $W_k$} \\
|
|
& \norm{x} & \text{the norm of the vector $\vec{x}$} \\
|
|
& [x]_\beta & \text{the coordinate vector of $x$ relative to $\beta$} \\
|
|
& \langle x, y \rangle & \text{the inner product of $\vec{x}$ and $\vec{y}$} \\
|
|
& \Z_2 & \text{the field consisting of $0$ and $1$} \\
|
|
& \overline{\vec{z}} & \text{the complex conjugate of $\vec{z}$} \\
|
|
& \vec{0} & \text{the zero vector} \\
|
|
\end{align*}
|