# 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: Another example: 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 1s on the main diagonal and 0s 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 = AT

## Hermitian

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

3
2+3i
−2i
5−i
2−3i
9
12
1+4i
2i
12
1
7
5+i
1−4i
7
12
Hermitian matrix

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 = AT

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

It is named after French mathematician Charles Hermite.