A Matrix Algebra

In this book, we reserve boldface letter to denote vectors (of scalars and random variables), and “blackboard bold” typeface to denote matrices.

We always write a vector as a column \[ \pmb{v}=\begin{bmatrix} v_1 \\ v_2 \\ \vdots \\ v_k \end{bmatrix}_{k\times1} \]

Definition A.1 (Transpose of a Matrix) Let \(\mathbb{A}_{k\times l}\) be a matrix, it’s transpose, denoted \(\mathbb{A}^T\), is an \(l\times k\) matrix such that the \((i,j)\)-th entry of \(\mathbb{A}\) becomes the \((j,i)\)-th entry of \(\mathbb{A}^T\).

Definition A.2 (Sum of Matrices) Let \(\mathbb{A},\mathbb{B}\) are matrices both of size \(k\times l\), then the sum \(\mathbb{A}+\mathbb{B}\) is defined as the another matrix \(\mathbb{C}\) size \(k\times l\) such that the \((i,j)\)-th entry is the sum of the \((i,j)\)-th entries of \(\mathbb A\) and \(\mathbb B\). \[ \mathbb{C}=\begin{bmatrix} a_{11}+b_{11} & a_{12}+b_{12} & \ldots & a_{1l}+b_{1l} \\ a_{21}+b_{21} & a_{22}+b_{22} & \ldots & a_{2l}+b_{2l} \\ \vdots & \vdots & \ddots & \vdots \\ a_{k1}+b_{k1} & a_{k2}+b_{k2} & \ldots & a_{kl}+b_{kl} \\ \end{bmatrix}_{k\times l} \]

Definition A.3 (Product of Matrices) Let \(\mathbb{A},\mathbb{B}\) are matrices both of size \(k\times l\), then the sum \(\mathbb{A}+\mathbb{B}\) is defined as the another matrix \(\mathbb{C}\) size \(k\times l\) such that the \((i,j)\)-th entry is the sum of the \((i,j)\)-th entries of \(\mathbb A\) and \(\mathbb B\). \[ \mathbb{C}=\begin{bmatrix} a_{11}+b_{11} & a_{12}+b_{12} & \ldots & a_{1l}+b_{1l} \\ a_{21}+b_{21} & a_{22}+b_{22} & \ldots & a_{2l}+b_{2l} \\ \vdots & \vdots & \ddots & \vdots \\ a_{k1}+b_{k1} & a_{k2}+b_{k2} & \ldots & a_{kl}+b_{kl} \\ \end{bmatrix}_{k\times l} \]