2. Matrices and Matrix Operations

Matrices and Matrix Operations

Matrix Representation of Linear Transformations

A linear transformation can be represented by a matrix. Let $\mathbf{A}$ be an $m \times n$ matrix and $\mathbf{x}$ be a vector in $\mathbb{R}^n$. The product $\mathbf{Ax}$ represents the transformation of $\mathbf{x}$ by $\mathbf{A}$.

Matrix Addition, Scalar Multiplication, and Multiplication

Matrix Addition: If $\mathbf{A}$ and $\mathbf{B}$ are $m \times n$ matrices, their sum $\mathbf{A} + \mathbf{B}$ is obtained by adding corresponding elements:

$$ \mathbf{A} + \mathbf{B} = \begin{bmatrix} a_{11} + b_{11} & a_{12} + b_{12} & \cdots & a_{1n} + b_{1n} \\ a_{21} + b_{21} & a_{22} + b_{22} & \cdots & a_{2n} + b_{2n} \\ \vdots & \vdots & \ddots & \vdots \\ a_{m1} + b_{m1} & a_{m2} + b_{m2} & \cdots & a_{mn} + b_{mn} \end{bmatrix} $$

Scalar Multiplication: If $c$ is a scalar, the product $c\mathbf{A}$ of the scalar $c$ and the matrix $\mathbf{A}$ is obtained by multiplying each element of $\mathbf{A}$ by $c$:

$$ c\mathbf{A} = \begin{bmatrix} ca_{11} & ca_{12} & \cdots & ca_{1n} \\ ca_{21} & ca_{22} & \cdots & ca_{2n} \\ \vdots & \vdots & \ddots & \vdots \\ ca_{m1} & ca_{m2} & \cdots & ca_{mn} \end{bmatrix} $$

Matrix Multiplication: If $\mathbf{A}$ is an $m \times p$ matrix and $\mathbf{B}$ is a $p \times n$ matrix, their product $\mathbf{AB}$ is an $m \times n$ matrix obtained by multiplying the rows of $\mathbf{A}$ by the columns of $\mathbf{B}$:

$$ \mathbf{AB} = \mathbf{A}\begin{bmatrix} | & | & & | \\ \mathbf{B}_1 & \mathbf{B}_2 & \cdots & \mathbf{B}_n \\ | & | & & | \end{bmatrix} = \begin{bmatrix} \mathbf{A}\mathbf{B}_1 & \mathbf{A}\mathbf{B}_2 & \cdots & \mathbf{A}\mathbf{B}_n \end{bmatrix} $$

where $\mathbf{B}_1, \mathbf{B}_2, \ldots, \mathbf{B}_n$ are the columns of $\mathbf{B}$.

Transpose, Inverse, and Determinant of Matrices

Transpose: The transpose of an $m \times n$ matrix $\mathbf{A}$, denoted by $\mathbf{A}^T$, is obtained by interchanging rows and columns:

$$ \mathbf{A}^T = \begin{bmatrix} a_{11} & a_{21} & \cdots & a_{m1} \\ a_{12} & a_{22} & \cdots & a_{m2} \\ \vdots & \vdots & \ddots & \vdots \\ a_{1n} & a_{2n} & \cdots & a_{mn} \end{bmatrix} $$

Inverse: If $\mathbf{A}$ is a square matrix, and if there exists a matrix $\mathbf{B}$ such that $\mathbf{AB} = \mathbf{BA} = \mathbf{I}$, then $\mathbf{B}$ is called the inverse of $\mathbf{A}$, denoted by $\mathbf{A}^{-1}$. Not all matrices have inverses.

Determinant: The determinant of a square matrix $\mathbf{A}$, denoted by $\det(\mathbf{A})$ or $|\mathbf{A}|$, is a scalar value calculated from the elements of the matrix. It represents the scaling factor of the transformation represented by the matrix.