Types of Matrix

First, some definitions!

A Matrix is an array of numbers:

2x3 Matrix
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:

main diagonal of a 3x3 matrix

Another example:

main diagonal of a 2x3 matrix

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

transpose of a matrix

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.

2x2 Matrix
A square matrix (2 rows, 2 columns)

3x3 Matrix
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 = A^T

Hermitian

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

2+3i

−2i

5−i

2−3i

1+4i

5+i

1−4i

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 = 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.

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