Revisit to Matrix Multiplication
Table of contents
- 1. inner product (traditional)
- 2. row operation, column operation
- 3. well-definedness
- 4. outer product
- +5. block multiplication
1. inner product (traditional)
\[AB = \begin{pmatrix} \mathbf{a}_{1}^{*} \\ \mathbf{a}_{2}^{*} \\ \mathbf{a}_{2}^{*} \end{pmatrix} \begin{pmatrix} \mathbf{b}_{1} & \mathbf{b}_{2} & \mathbf{b}_{3} \end{pmatrix} = \bigg( \mathbf{a}_{i}^{*} \cdot \mathbf{b}_{j} \bigg)_{i \times j}\]2. row operation, column operation
\[\begin{align*} EB & = E \begin{pmatrix} \mathbf{b}_1^* \\ \mathbf{b}_2^* \\ \mathbf{b}_3^* \end{pmatrix} \text{ : row operation (left multiplication)} \\ & \overset{\text{ex}}{=} \begin{pmatrix} 1 & 0 & 0 \\ 0 & 1 & 0 \\ -l& 0 & 1 \end{pmatrix} \begin{pmatrix} \mathbf{b}_1^* \\ \mathbf{b}_2^* \\ \mathbf{b}_3^* \end{pmatrix} = \begin{pmatrix} \mathbf{b}_1^* \\ \mathbf{b}_2^* \\ -l\mathbf{b}_1^* + \mathbf{b}_3^* \end{pmatrix} \\ \\ AF &= \begin{pmatrix} \mathbf{a}_{1} & \mathbf{a}_{2} & \mathbf{a}_{3} \end{pmatrix} F \text{ : column operaion (right multiplication)} \\ & \overset{\text{ex}}{=} \begin{pmatrix} \mathbf{a}_{1} & \mathbf{a}_{2} & \mathbf{a}_{3} \end{pmatrix} \begin{pmatrix} 1 & 0 & 0 \\ 0 & 1 & 0 \\ -l& 0 & 1 \end{pmatrix} = \begin{pmatrix} \mathbf{a}_{1} - l\mathbf{a}_3 & \mathbf{a}_{2} & \mathbf{a}_{3} \end{pmatrix} \end{align*}\]3. well-definedness
\[\begin{align*} AB &= A \begin{pmatrix} \mathbf{b}_{1} & \mathbf{b}_{2} & \mathbf{b}_{3} \end{pmatrix} = \begin{pmatrix} A\mathbf{b}_{1} & A\mathbf{b}_{2} & A\mathbf{b}_{3} \end{pmatrix} \\ AB &= \begin{pmatrix} \mathbf{a}_{1}^{*} \\ \mathbf{a}_{2}^{*} \\ \mathbf{a}_{2}^{*} \end{pmatrix} B = \begin{pmatrix} \mathbf{a}_{1}^{*}B \\ \mathbf{a}_{2}^{*}B \\ \mathbf{a}_{2}^{*}B \end{pmatrix} \end{align*}\]4. outer product
\[AB = \begin{pmatrix} \mathbf{a}_{1} & \mathbf{a}_{2} & \mathbf{a}_{3} \end{pmatrix} \begin{pmatrix} \mathbf{b}_1^* \\ \mathbf{b}_2^* \\ \mathbf{b}_3^* \end{pmatrix} = \mathbf{a_1}\mathbf{b_1^*} + \mathbf{a_2}\mathbf{b_2^*} + \mathbf{a_3}\mathbf{b_3^*}\] \[A\mathbf{x} = \begin{bmatrix} \mathbf{a_1} \ \mathbf{a_2} \ \cdots \ \mathbf{a_n} \end{bmatrix} \begin{bmatrix} x_1 \\ x_2 \\ \vdots \\ x_n \end{bmatrix} = \mathbf{a_1}x_1 + \mathbf{a_2}x_2 + \cdots + \mathbf{a_n}x_n \in C(A) \\ \implies \exists \mathbf{x} (A\mathbf{x}=\mathbf{b}) \iff \mathbf{b} \in C(A)\]+5. block multiplication
: exactly same as general multiplication