Section04 Rules for Matrix Operations

Perspectives of $\boldsymbol{AB}$

From Dot Product

$$\begin{array}{c} \left\{ \begin{array}{ll} \boldsymbol{A} = \left[ \begin{matrix} \boldsymbol{a}_{1 \boldsymbol{\cdot}} \\ \boldsymbol{a}_{2 \boldsymbol{\cdot}} \\ \vdots \\ \boldsymbol{a}_{m \boldsymbol{\cdot}} \end{matrix} \right] & (\boldsymbol{a}_{i \boldsymbol{\cdot}} \text{ is } i^{\text{th}} \text{ row of Matrix } \boldsymbol{A_{m \times n}}) \\\\ \boldsymbol{B} = \left[ \begin{matrix} \boldsymbol{b}_{\boldsymbol{\cdot}1} & \boldsymbol{b}_{\boldsymbol{\cdot}2} & \cdots & \boldsymbol{b}_{\boldsymbol{\cdot}p} \end{matrix} \right] & (\boldsymbol{b}_{\boldsymbol{\cdot}j} \text{ is } j^{\text{th}} \text{ column of Matrix } \boldsymbol{B_{n \times p}} ) \end{array} \right.\\\\ \big\Downarrow \\\\ \boldsymbol{AB} = \left[ \begin{matrix} \boldsymbol{a}_{1\boldsymbol{\cdot}} \cdot \boldsymbol{b}_{\boldsymbol{.} 1} & \boldsymbol{a}_{1\boldsymbol{\cdot}} \cdot \boldsymbol{b}_{\boldsymbol{.} 2} & \cdots & \boldsymbol{a}_{1\boldsymbol{\cdot}} \cdot \boldsymbol{b}_{\boldsymbol{.} p} \\ \boldsymbol{a}_{2\boldsymbol{\cdot}} \cdot \boldsymbol{b}_{\boldsymbol{.} 1} & \boldsymbol{a}_{2\boldsymbol{\cdot}} \cdot \boldsymbol{b}_{\boldsymbol{.} 2} & \cdots & \boldsymbol{a}_{2\boldsymbol{\cdot}} \cdot \boldsymbol{b}_{\boldsymbol{.} p} \\ \vdots & \vdots & \ddots & \vdots \\ \boldsymbol{a}_{m\boldsymbol{\cdot}} \cdot \boldsymbol{b}_{\boldsymbol{.} 1} & \boldsymbol{a}_{m\boldsymbol{\cdot}} \cdot \boldsymbol{b}_{\boldsymbol{.} 2} & \cdots & \boldsymbol{a}_{m\boldsymbol{\cdot}} \cdot \boldsymbol{b}_{\boldsymbol{.} p} \\ \end{matrix} \right] = \boldsymbol{C}_{m \times p}\\ \big\Downarrow\\ c_{ij} = \boldsymbol{a}_{i \boldsymbol{\cdot}} \cdot \boldsymbol{b}_{\boldsymbol{\cdot}j} \end{array}$$

Outer product

$$row \cdot col$$

Inner product

$$col \cdot row$$

Matrix $\boldsymbol{A}$ times every columns of $\boldsymbol{B}$

$$\boldsymbol{AB} = \left[ \begin{matrix} \boldsymbol{Ab}_{\boldsymbol{\cdot}1} & \boldsymbol{Ab}_{\boldsymbol{\cdot}2} & \cdots & \boldsymbol{Ab}_{\boldsymbol{\cdot}p} \end{matrix} \right] = \left[ \begin{matrix} \boldsymbol{\boldsymbol{c}_{\boldsymbol{\cdot}1}} & \boldsymbol{\boldsymbol{c}_{\boldsymbol{\cdot}2}} & \cdots & \boldsymbol{\boldsymbol{c}_{\boldsymbol{\cdot}p}} \end{matrix} \right]$$

Every row of $\boldsymbol{A}$ times matrix $\boldsymbol{B}$

$$\boldsymbol{AB} = \left[ \begin{matrix} \boldsymbol{a}_{1 \boldsymbol{\cdot}}\boldsymbol{B} \\ \boldsymbol{a}_{2 \boldsymbol{\cdot}}\boldsymbol{B} \\ \vdots \\ \boldsymbol{a}_{m \boldsymbol{\cdot}}\boldsymbol{B} \end{matrix} \right] = \left[ \begin{matrix} \boldsymbol{c}_{1 \boldsymbol{\cdot}} \\ \boldsymbol{c}_{2 \boldsymbol{\cdot}} \\ \vdots \\ \boldsymbol{c}_{m \boldsymbol{\cdot}} \end{matrix} \right]$$

Columns Multiply Rows

$$\boldsymbol{AB} = \left[ \begin{matrix} \boldsymbol{a}_{\boldsymbol{\cdot} 1} & \boldsymbol{a}_{\boldsymbol{\cdot} 2} & \cdots & \boldsymbol{a}_{\boldsymbol{\cdot} n} \end{matrix} \right] \left[ \begin{matrix} \boldsymbol{b}_{1 \boldsymbol{\cdot}} \\ \boldsymbol{b}_{2 \boldsymbol{\cdot}} \\ \vdots \\ \boldsymbol{b}_{n \boldsymbol{\cdot}} \end{matrix} \right] = \sum_{k=1}^{n}{\boldsymbol{a}_{\boldsymbol{\cdot}k}\boldsymbol{b}_{k \boldsymbol{\cdot}}}$$

The Laws for Matrix Operations

Commutiative law

$$\boldsymbol{A} + \boldsymbol{B} = \boldsymbol{B} + \boldsymbol{A}\\ \color{#D0104C}{\boldsymbol{AB} \ne \boldsymbol{BA}} \\ \color{#D0104C}{\boldsymbol{AI} = \boldsymbol{IA}}$$

Distributive Law

$$c(\boldsymbol{A} + \boldsymbol{B}) = c \boldsymbol{A} + c \boldsymbol{B} \\ \boldsymbol{A}(\boldsymbol{B} + \boldsymbol{C}) = \boldsymbol{AB} + \boldsymbol{AC} \\ (\boldsymbol{A} + \boldsymbol{B}) \boldsymbol{C} = \boldsymbol{AC} + \boldsymbol{BC}$$

Associative Law

$$\boldsymbol{A} + (\boldsymbol{B} + \boldsymbol{C}) = (\boldsymbol{A} + \boldsymbol{B}) + \boldsymbol{C} \\ \boldsymbol{A}(\boldsymbol{BC}) = (\boldsymbol{AB}) \boldsymbol{C}$$

Others

$$\begin{array}{c} \text{When Matrix } \boldsymbol{A} \text{ is a square.} \boldsymbol{AA} \text{ can be represented by } \boldsymbol{A}^{2}\\ (\boldsymbol{A}^{p})(\boldsymbol{A}^{q}) = \boldsymbol{A}^{p+q} \\ (\boldsymbol{A}^{p})^{q} = \boldsymbol{A}^{pq} \end{array}$$

Block Matrices and Block Multiplication

Block Multiplication

• If blocks of $\boldsymbol{A}$ can multiply blocks of $\boldsymbol{B}$ $$\left[ \begin{matrix} A_{11} & A_{12} \\ A_{21} & A_{22} \\ \end{matrix} \right] \left[ \begin{matrix} B_{11} \\ B_{21} \end{matrix} \right] = \left[ \begin{matrix} A_{11}B_{11} + A_{12}B_{21} \\ A_{21}B_{11} + A_{22}B_{21} \\ \end{matrix} \right]$$

Elimination by Blocks

$$\begin{array}{c} \begin{array}{ll} \boldsymbol{E} = \left[ \begin{array}{@{}c|c@{}} \begin{matrix} \boldsymbol{I} \end{matrix} & \boldsymbol{0} \\ \hline -\boldsymbol{CA}^{-1} & \begin{matrix} \boldsymbol{I} \end{matrix} \end{array} \right] & \boldsymbol{M} = \left[ \begin{array}{@{}c|c@{}} \begin{matrix} \boldsymbol{A} \end{matrix} & \boldsymbol{B} \\ \hline \boldsymbol{C} & \begin{matrix} \boldsymbol{D} \end{matrix} \end{array} \right]\\ \end{array} \\\\ \boldsymbol{EM} = \left[ \begin{matrix} \boldsymbol{I} \boldsymbol{A} & \boldsymbol{I} \boldsymbol{B} \\ (-\boldsymbol{CA}^{-1})\boldsymbol{A} + \boldsymbol{IC} & -\boldsymbol{CA}^{-1}\boldsymbol{B} + \boldsymbol{ID} \end{matrix} \right] = \left[ \begin{array}{@{}c|c@{}} \begin{matrix} \boldsymbol{A} \end{matrix} & \boldsymbol{B} \\ \hline \boldsymbol{0} & \begin{matrix} \boldsymbol{D} - \boldsymbol{CA}^{-1}\boldsymbol{B} \end{matrix} \end{array} \right] \end{array}$$

Schur complement

$$\boldsymbol{D} - \boldsymbol{CA}^{-1}\boldsymbol{B}$$

Properties from Exercise

adjacency matrix

• example
  • graph
  • represented by Matrix ( Adjacency matrix ) $$\begin{array}{c} \boldsymbol{S} = \left[ \begin{matrix} 0 & 1 & 1 & 0 \\ 1 & 0 & 1 & 1 \\ 1 & 1 & 0 & 1 \\ 0 & 1 & 1 & 0 \end{matrix} \right] \\ s_{ij} = \left\{ \begin{array}{ll} 0 & \text{if there isn't a path between node } i \text{ and node }j. \\ 1 & \text{if there is a path between node } i \text{ and node }j. \end{array} \right. \end{array}$$
  • the meaning of $\boldsymbol{S}^{n}$ $$( \boldsymbol{S}^{n} )_{ij} \text{ counts the walks of length } n \text{ between node } i \text{ and node } j.$$

Entry in row 1, column of 1 of $\boldsymbol{CDE}$

$$\begin{array}{c} \begin{array}{ll} \because & (\boldsymbol{CDE})_{11} = (\boldsymbol{CD})_{1 \boldsymbol{\cdot}} \boldsymbol{e}_{\boldsymbol{\cdot} 1}\\ & (\boldsymbol{CD})_{1 \boldsymbol{\cdot}} = \boldsymbol{c}_{1 \boldsymbol{\cdot}}\boldsymbol{D} \\ \therefore & (\boldsymbol{CDE})_{11} = \boldsymbol{c}_{1 \boldsymbol{\cdot}}\boldsymbol{D}\boldsymbol{e}_{\boldsymbol{\cdot} 1} \\ \end{array} \\ \Downarrow \\ (\boldsymbol{CDE})_{ij} = (\text{row } i \text{ of } \boldsymbol{C})\boldsymbol{D}(\text{col } j \text{ of } \boldsymbol{E}) = \boldsymbol{c}_{i \boldsymbol{\cdot}}\boldsymbol{D}\boldsymbol{e}_{\boldsymbol{\cdot} j} \end{array}$$