Matrix-Class

A set of class methods used for common matrix operations

Addition / Subtraction

Matrix addition and subtraction is an element by element operation. Two matrices must have the same dimensions in order to be added or subtracted.

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

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

Scalar Multiplication

When multiplying a matrix $\mathbf{A}$ by a scalar $c$, all of the entries in $\mathbf{A}$ are multiplied by $c$:

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

Matrix Multiplication

Multiplication of Matrix $\mathbf{A}$ with matrix $\mathbf{B}$ is only possible if the width of $\mathbf{A}$ is equal to the height of $\mathbf{B}$

If $\mathbf{A}$ is an $m \times n$ matrix and $\mathbf{B}$ is an $n \times p$ matrix, their product $\mathbf{AB}$ is an $m \times p$ matrix.

When multiplying two matrices, we can calculate the value of the element at row $i$ and column $j$ with the following equation:

$$(\mathbf{AB})_{ij} = \sum_{k=1}^n a_{ik}b_{kj}$$

Transpose

The transpose of a matrix $\mathbf{A}$ is given by $\mathbf{A^T}$ and can be thought of in several ways:

• The rows of $\mathbf{A^T}$ are the columns of $\mathbf{A}$.
• The columns of $\mathbf{A^T}$ are the rows of $\mathbf{A}$.

Mathematically, the element at row $i$ and column $j$ of the transpose is given by:

$$[\mathbf{A^T}]_{ij} = [\mathbf{A}]_{ji}$$

Trace

The trace of an $n\times n$ square matrix $\mathbf{A}$ is the sum of the elements on the main diagonal of the matrix.

$$\text{tr}\left(\mathbf{A}\right) = \sum_{i=1}^n a_{ii} = a_{11} + a_{22} + \dots + a_{nn}$$

Determinant

The determinant is a useful value when describing a matrix. It can be denoted in one of three ways:

1. $\text{det } \left(\mathbf{A}\right)$
2. $\text{det } \mathbf{A}$
3. $|\mathbf{A}|$

1x1 Matrices

The determinant of a $1\times1$ matrix is just the value of the matrice's only element. For example if $\mathbf{A} = \begin{vmatrix}4\end{vmatrix}$, then the determinant of $\mathbf{A}$ is given by:

$$\begin{vmatrix}\mathbf{A}\end{vmatrix} = 4$$

2x2 Matrices

The determinant of a $2\times2$ matrix is given by:

$$\begin{vmatrix}\mathbf{A}\end{vmatrix} = \begin{vmatrix} a & b \\ c & d \end{vmatrix} = ad - bc $$

Larger Matrices

If you are interested in learning more you should look at the Wikipedia article: Determinant.

Inverse

The inverse of a matrix $\mathbf{A}$ is given by $\mathbf{A^{-1}}$

A matrix $\mathbf{A}$ is invertible if there exists a matrix $\mathbf{B}$ such that the product of $\mathbf{A}$ and $\mathbf{B}$ is the identity matrix $\mathbf{I}$:

$$\mathbf{AB} = \mathbf{BA} = \mathbf{I}$$

1x1 Matrices

For a $1\times1$ matrix with a single element with value $a$, the inverse is simlpy $\frac{1}{a}$

2x2 Matrices

The inverse of a $2\times 2$ matrix is given by the following equation:

$$\mathbf{A}^{-1} = \frac{1}{\text{det }\mathbf{A}} \left[\left(\text{tr } \mathbf{A}\right) \mathbf{I} - \mathbf{A}\right]$$

Larger Matrices

If you are interested in learning more you should look at the Wikipedia article Invertible Matrix.