## 6.4 Basic matrix arthimetic

Some basic rules, for suitably-sized matrices $$A$$ and $$B$$:

• $$A + B = B + A$$.
• $$c A$$ for some constant $$c$$ multiplies every element of matrix $$A$$ by $$c$$.
• $$A B \ne B A$$ in general.
• $$(A B)^T = B^T A^T$$.
• $$(A B)^{-1} = B^{-1} A^{-1}$$.
• $$I A = A I = A$$ for a suitably-sized identity matrix $$I$$.
• If $$A$$ is symmetric, then $$A^{-1}$$ (if it exists) is also symmetric.

Two specific cases sometimes come in handy. Consider a vector $$\textbf{y}$$ of length $$n$$ (a matrix of size $$n\times 1$$): $\textbf{y} = \left[ \begin{array}{c} y_1\\ y_2\\ y_3\\ \vdots\\ y_n \end{array} \right].$ Then multiplying $$\textbf{y}^T$$ by $$\textbf{y}$$ produces a $$1\times 1$$ vector (a scalar): \begin{aligned} \textbf{y}^T\textbf{y} &= \left[ \begin{array}{ccccc} y_1\quad& y_2\quad& y_3\quad& \cdots\quad& y_n \end{array} \right] \times \left[ \begin{array}{c} y_1\\ y_2\\ y_3\\ \vdots\\ y_n \end{array} \right]\\ &= y_1^2 + y_2^2 + \cdots + y_n^2 = \sum_{i=1}^n y_i. \end{aligned} Also, \begin{aligned} \textbf{y}\textbf{y}^T &= \left[ \begin{array}{c} y_1\\ y_2\\ y_3\\ \vdots\\ y_n \end{array} \right] \times \left[ \begin{array}{ccccc} y_1\quad& y_2\quad& y_3\quad& \cdots\quad& y_n \end{array} \right]\\ &= \left[ \begin{array}{cccc} y_1^2 & y_1 y_2 & \dots & y_1 y_n \\ y_2 y_1 & y_2^2 & \dots & y_2 y_n\\ \vdots & \vdots & \ddots & \vdots\\ y_n y_1 & y_n y_2 & \dots & y_n^2 \end{array} \right]. \end{aligned}