Matrices

Matrices are useful for writing linear systems more compactly. Instead of writing linear systems as:

We could write the linear systems using matrices:

$$\begin{pmatrix} 1 & 2 \\\\ 3 & 4 \\\\ 5 & 6 \end{pmatrix} \begin{pmatrix} x_{1} \\\\ x_{2} \end{pmatrix} = \begin{pmatrix} -1 \\\\ 0 \\\\ 5 \end{pmatrix}$$

Matrix calculations

The four relevant operators are:

1. Addition
2. Subtraction
3. Scalar multiplication
4. Matrix multiplication

Addition and subtraction are straight-forward and work only with matrices that have the same dimensions. Scalar multiplication is also straight-forward; each element would be multiplied by a single number. Matrix multiplication, takes each row of the left factor and multiplies it with each column of the right factor. For example:

$$\begin{pmatrix} 1 & 2 \\\\ 3 & 4 \\\\ 5 & 6 \end{pmatrix} \begin{pmatrix} x_{1} & y_{1} \\\\ x_{2} & y_{2} \end{pmatrix} = \begin{pmatrix} 1x_{1} + 2x_{2} & 1y_{1} + 2y_{2} \\\\ 3x_{1} + 4x_{2} & 3y_{1} + 4y_{2} \\\\ 5x_{1} + 6x_{2} & 5y_{1} + 6y_{2} \end{pmatrix}$$

An $m \times n$ matrix multiplied by an $n \times p$ matrix, yields an $m \times p$ matrix. In the example above, the multiplication of a 3 by 2 matrix to a 2 by 2 matrix, yielded a 3 by 2 matrix. Furthermore, changing the order of factors usually results in a completely different product and matrices can only be multiplied if the number of columns in the left factor matches the number of row in the right factor.

Special matrices

A zero matrix is a matrix where all elements are equal to zero.

$$\begin{pmatrix} 0 & 0 \\\\ 0 & 0 \end{pmatrix}$$

If we transpose this 2 by 3 matrix:

$$\begin{pmatrix} 1 & 2 & 3 \\\\ 4 & 5 & 6 \end{pmatrix}$$

we get this 3 by 2 matrix:

$$\begin{pmatrix} 1 & 4 \\\\ 2 & 5 \\\\ 3 & 6 \end{pmatrix}$$

Symmetric matrices are square matrices that are symmetric around their main diagonals; a symmetric matrix is always equal to its transpose. For example:

$$\begin{pmatrix} 1 & 1 & 1 & 1 \\\\ 1 & 2 & 4 & 4 \\\\ 1 & 4 & 3 & 9 \\\\ 1 & 4 & 9 & 4 \end{pmatrix}$$

Triangular matrices are square matrices in which the elements either above or below the main diagonal are all equal to zero. Here’s an example of an upper triangular matrix:

$$\begin{pmatrix} 1 & 1 & 1 & 1 \\\\ 0 & 2 & 4 & 4 \\\\ 0 & 0 & 3 & 9 \\\\ 0 & 0 & 0 & 4 \end{pmatrix}$$

Here’s an example of a lower triangular matrix:

$$\begin{pmatrix} 1 & 0 & 0 & 0 \\\\ 1 & 2 & 0 & 0 \\\\ 1 & 4 & 3 & 0 \\\\ 1 & 4 & 9 & 4 \end{pmatrix}$$

A diagonal matrix is a square matrix in which all elements that are not part of its main diagonal are equal to zero. For example:

$$\begin{pmatrix} 1 & 0 & 0 & 0 \\\\ 0 & 2 & 0 & 0 \\\\ 0 & 0 & 3 & 0 \\\\ 0 & 0 & 0 & 4 \end{pmatrix}$$

Identity matrices are are square matrices with $n $ rows, where all elements on the main diagonal are equal to 1 and all other elements are 0. For example:

$$\begin{pmatrix} 1 & 0 & 0 & 0 \\\\ 0 & 1 & 0 & 0 \\\\ 0 & 0 & 1 & 0 \\\\ 0 & 0 & 0 & 1 \end{pmatrix}$$

Multiplying a matrix with the identity matrix yields a product equal to the original matrix.

If the product of two square matrices is an identity matrix, then the two factor matrices are inverses of each other. For example:

$$\begin{pmatrix} 1 & 2 \\\\ 3 & 4 \end{pmatrix} \times \begin{pmatrix} x_{11} & x_{12} \\\\ x_{21} & x_{22} \end{pmatrix} = \begin{pmatrix} 1 & 0 \\\\ 0 & 1 \end{pmatrix}$$