# Types of Matrix

First, some definitions!

A Matrix is an array of numbers:

A Matrix

(This one has 2 Rows and 3 Columns)

We talk about one **matrix**, or several **matrices**.

The **Main Diagonal** starts at the top left and goes down to the right:

A **Transpose** is where we swap entries across the main diagonal (rows become columns) like this:

The main diagonal stays the same.

Here are some of the most common types of matrix:

## Square

A **square** matrix has the same number of rows as columns.

A square matrix (2 rows, 2 columns)

Also a square matrix (3 rows, 3 columns)

## Identity Matrix

An **Identity Matrix** has **1**s on the main diagonal and **0**s everywhere else:

A 3×3 Identity Matrix

- It is square (same number of rows as columns)
- It can be large or small (2×2, 100×100, ... whatever)
- Its symbol is the capital letter
**I**

It is the matrix equivalent of the number "1", when we multiply with it the original is unchanged:

A × I = A

I × A = A

## Diagonal Matrix

A diagonal matrix has zero anywhere not on the main diagonal:

A diagonal matrix

## Scalar Matrix

A scalar matrix has all main diagonal entries the same, with zero everywhere else:

A scalar matrix

## Triangular Matrix

**Lower triangular** is when all entries above the main diagonal are zero:

A lower triangular matrix

**Upper triangular** is when all entries below the main diagonal are zero:

An upper triangular matrix

## Zero Matrix (Null Matrix)

Zeros just everywhere:

Zero matrix

## Symmetric

In a Symmetric matrix matching entries either side of the main diagonal are **equal**, like this:

Symmetric matrix

It must be square, and is equal to its own transpose

A = A^{T}

## Hermitian

A Hermitian matrix is symmetric except for the imaginary parts that swap sign across the main diagonal:

See how **+i** changes to **−i** and vice versa?

Changing the sign of the second part is called the conjugate, and so the correct definition is:

A Hermitian matrix is equal to its own **conjugate transpose**:

A = A^{T}

This also means the main diagonal entries must be purely real (to be their own conjugate).

It is named after French mathematician Charles Hermite.