\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 $A$} \\ & \mathsf{C}^n(\R) & \text{set of functions $f$ on $\R$ with $f^{(n)}$ continuous} \\ & \mathsf{C}^\infty & \text{set of functions with derivatives of every order} \\ & \mathsf{C}(\R) & \text{the vector space of continuous functions on $\R$} \\ & \mathsf{C}([0,1]) & \text{the vector space of continuous functions on $[0,1]$} \\ & \mathsf{C}_x & \text{the $T$-cyclic subspaces generated by $x$} \\ & \mathsf{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 $\F$} \\ & \mathsf{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*}