Subsection 4.6.2 Summary
¶Subsubsection 4.6.2.1 Partitioned matrix-vector multiplication}
¶
\begin{equation*}
\begin{array}{rcl}
\left( \begin{array}{ c | c | c | c }
A_{0,0} \amp A_{0,1} \amp \cdots \amp A_{0,N-1} \\ \hline
A_{1,0} \amp A_{1,1} \amp \cdots \amp A_{1,N-1} \\ \hline
\vdots \amp \vdots \amp \ddots \amp \vdots \\ \hline
A_{M-1,0} \amp A_{M-1,1} \amp \cdots \amp A_{M-1,N-1}
\end{array}
\right)
\left( \begin{array}{ c }
x_0 \\ \hline
x_1 \\ \hline
\vdots \\ \hline
x_{N-1}
\end{array}
\right)
=
\begin{MatrixC}
A_{0,0} x_{0} + A_{0,1} x_{1} + \cdots + A_{0,N-1} x_{N-1} \\ \hline
A_{1,0} x_{0} + A_{1,1} x_{1} + \cdots + A_{1,N-1} x_{N-1} \\ \hline
\vdots \\ \hline
A_{M-1,0} x_{0} + A_{M-1,1} x_{1} + \cdots + A_{M-1,N-1} x_{N-1}
\end{MatrixC}.
\end{array}
\end{equation*}
Subsubsection 4.6.2.2 Transposing a partitioned matrix}
¶
\begin{equation*}
\left( \begin{array}{c | c | c | c}
A_{0,0} \amp A_{0,1} \amp \cdots \amp A_{0,N-1} \\ \hline
A_{1,0} \amp A_{1,1} \amp \cdots \amp A_{1,N-1} \\ \hline
\vdots \amp \vdots \amp \amp \vdots \\ \hline
A_{M-1,0} \amp A_{M-1,1} \amp \cdots \amp A_{M-1,N-1}
\end{array}
\right)^T
=
\left( \begin{array}{c | c | c | c}
A_{0,0}^T \amp A_{1,0}^T \amp \cdots \amp A_{M-1,0}^T \\ \hline
A_{0,1}^T \amp A_{1,1}^T \amp \cdots \amp A_{M-1,1}^T \\ \hline
\vdots \amp \vdots \amp \amp \vdots \\ \hline
A_{0,N-1}^T \amp A_{1,N-1}^T \amp \cdots \amp A_{M-1,N-1}^T
\end{array}
\right).
\end{equation*}
Subsubsection 4.6.2.3 Composing linear transformations
¶Let \(L_A: \mathbb{R}^k \rightarrow \mathbb{R}^m \) and \(L_B: \mathbb{R}^n \rightarrow \mathbb{R}^k \) both be linear transformations and, for all \(x \in \mathbb{R}^n \text{,}\) define the function \(L_C: \mathbb{R}^n \rightarrow \mathbb{R}^m \) by \(L_C( x ) = L_A( L_B( x ) ) \text{.}\) Then \(L_C(x) \) is a linear transformations.
Subsubsection 4.6.2.4 Matrix-matrix multiplication
¶
\begin{equation*}
A B =
A
\left( \begin{array}{c | c | c | c }
b_0 \amp b_1 \amp \cdots \amp b_{n-1}
\end{array}
\right)
=
\left( \begin{array}{c | c | c | c }
A b_0 \amp A b_1 \amp \cdots \amp A b_{n-1}
\end{array}
\right).
\end{equation*}
If
\begin{equation*}
\begin{array}{c c}
C =
\left(
\begin{array}{c c c c }
\gamma_{0,0} \amp \gamma_{0,1} \amp \cdots \amp \gamma_{0,n-1} \\
\gamma_{1,0} \amp \gamma_{1,1} \amp \cdots \amp \gamma_{1,n-1} \\
\vdots \amp \vdots \amp \vdots \amp \vdots \\
\gamma_{m-1,0} \amp \gamma_{m-1,1} \amp \cdots \amp \gamma_{m-1,n-1} \\
\end{array}
\right)
,
\quad
A =
\left(
\begin{array}{c c c c }
\alpha_{0,0} \amp \alpha_{0,1} \amp \cdots \amp \alpha_{0,k-1} \\
\alpha_{1,0} \amp \alpha_{1,1} \amp \cdots \amp \alpha_{1,k-1} \\
\vdots \amp \vdots \amp \vdots \amp \vdots \\
\alpha_{m-1,0} \amp \alpha_{m-1,1} \amp \cdots \amp \alpha_{m-1,k-1} \\
\end{array}
\right), \\
\mbox{and}
\quad
B =
\left(
\begin{array}{c c c c }
\beta_{0,0} \amp \beta_{0,1} \amp \cdots \amp \beta_{0,n-1} \\
\beta_{1,0} \amp \beta_{1,1} \amp \cdots \amp \beta_{1,n-1} \\
\vdots \amp \vdots \amp \vdots \amp \vdots \\
\beta_{k-1,0} \amp \beta_{k-1,1} \amp \cdots \amp \beta_{k-1,n-1} \\
\end{array}
\right).
\end{array}
\end{equation*}
then \(C = A B \) means that \(\gamma_{i,j} = \sum_{p=0}^{k-1} \alpha_{i,p} \beta_{p,j} \text{.}\)
A table of matrix-matrix multiplications with matrices of special shape is given at the end of this week.
Subsubsection 4.6.2.5 Outer product
¶Let \(x \in \mathbb{R}^m \) and \(y \in \mathbb{R}^n \text{.}\) Then the outer product of \(x \) and \(y \) is given by \(x y^T \text{.}\) Notice that this yields an \(m \times n \) matrix:
\begin{equation*}
\begin{array}{rcl}
x y^T \amp =\amp
\left( \begin{array}{c}
\chi_0 \\
\chi_1 \\
\vdots \\
\chi_{m-1}
\end{array} \right)
\left( \begin{array}{c}
\psi_0 \\
\psi_1 \\
\vdots \\
\psi_{n-1}
\end{array} \right)^T
=
\left( \begin{array}{c}
\chi_0 \\
\chi_1 \\
\vdots \\
\chi_{m-1}
\end{array} \right)
\left( \begin{array}{c c c c}
\psi_0 \amp
\psi_1 \amp
\cdots \amp
\psi_{n-1}
\end{array} \right)
\\
\amp =\amp
\left( \begin{array}{c c c c}
\chi_0 \psi_0 \amp \chi_0 \psi_1 \amp \cdots \amp \chi_0 \psi_{n-1} \\
\chi_1 \psi_0 \amp \chi_1 \psi_1 \amp \cdots \amp \chi_1 \psi_{n-1} \\
\vdots \amp \vdots \amp \amp \vdots \\
\chi_{m-1} \psi_0 \amp \chi_{m-1} \psi_1 \amp \cdots \amp \chi_{m-1} \psi_{n-1}
\end{array} \right).
\end{array}
\end{equation*}