# Normal matrix

On this post you will find what a normal matrix is, as well as examples of normal matrices. In addition, you will find the properties of this type of matrix and exercises solved step by step.

## What is a normal matrix?

The definition of normal matrix is as follows:

A normal matrix is a complex matrix that multiplied by its conjugate transpose matrix is equal to the product of the conjugate transpose by itself. That is, all normal matrices meet the following condition:

Where AH is the conjugate transpose matrix of A.

However, in the real numbers field the previous condition is the same as saying that a matrix commutes with its transpose, that is:

Since the conjugate transpose matrix of a real matrix is simply the transpose of the matrix.

## Examples of normal matrices

Once we have seen the meaning of normal matrix, let’s see some examples of this type of matrix to fully understand the concept:

### Example of a normal matrix with complex numbers

The following complex square matrix of dimension 2×2 is normal:

Below there is the demonstration of the normality of the matrix:

### Example of a normal matrix with real numbers

The following square matrix with real numbers of order 2 is also normal:

In this case, as it only has real numbers, to show that it is normal we just have to check that the matrix commutes with its transpose:

## Properties of normal matrices

Normal matrices have the following characteristics:

• Every normal matrix is diagonalizable.
• A skew-Hermitian matrix is a normal matrix.
• If A is a normal matrix, the eigenvalues of the conjugate transpose matrix AH are the conjugate eigenvalues of A.

• The eigenvectors of any normal matrix associated with different eigenvalues are orthogonal.
• Lastly, any orthogonal matrix formed by real numbers is also a normal matrix.

## Solved exercises of normal matrices

### Exercise 1

Check that the following complex matrix of order 2 is normal:

To prove that the matrix is normal we must first calculate its conjugate transpose:

And now we do the check by multiplying matrix A by matrix AH in the two possible orders:

The result of both multiplications is equal, thus, matrix A is normal.

### Exercise 2

Show that the following square real matrix of order 2 is normal:

In this problem the matrix has only real numbers, so it is enough to check that the matrix product between matrix A and its tranpose is the same regardless of the order of the multiplication:

The result of both products is the same, therefore, matrix A is normal.

### Exercise 3

Determine whether the following matrix of complex numbers is normal:

To verify that the matrix is normal we must first calculate its conjugate transpose:

And now we check whether matrix A and its conjugate transpose commutate:

The result of both multiplications is the same, so matrix A is normal.

### Exercise 4

Verify that the following 3×3 dimension real matrix is normal:

As the matrix is completely made up of real elements, it is sufficient to check that the matrix product between matrix A and its transpose is independent of the multiplication order:

The result of both products is identical, so matrix A is normal.

### Exercise 5

Determine if the following complex matrix of order 3 is normal:

First, we calculate the conjugate transpose of the matrix:

Now we should do the matrix multiplications between matrix A and its conjugate transpose in the two possible orders. However, the conjugate transpose matrix of A is equal to the matrix A itself, therefore this is a Hermitian matrix. And consequently, from the properties of normal matrices it follows that A is a normal matrix, because any Hermitian matrix is a normal matrix.

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