Normal Matrix

A matrix A is said to be a Normal matrix if the pre and post matrix multiplication of conjugate transpose of A with the matrix A is commutative. In other words, normal matrices are those matrices whose matrix multiplication with its own conjugate transpose is commutative.

Normal Matrix is a generalisation of Unitary, Hermitian, skew-Hermitian as well as symmetric and skew-symmetric matrices.

A normal matrix is also unitarily diagonalizable, which means to diagonalize a normal matrix the required modal matrix is a unitary matrix. Let us define a normal matrix more formally in mathematical terms.

Definition of Normal Matrix

A square matrix A of order n×n of complex numbers, that is A ∈ C^n×n is said to be a normal matrix if A^HA = AA^H where A^H is the conjugate transpose of A.

The matrix A is also said to be a Normal matrix if there exists a Unitary matrix U such that U acts as a modal matrix which diagonalizes A.

Examples of Normal Matrix

Let us take an example of a complex matrix of order 2 × 2,

So according to the definition of normal matrix let us find the conjugate transpose of the given matrix.

Let

Then, A^H = (conjugate of A)^T =

Then A^HA =

And AA^H =

Hence A^HA = AA^H, that is a normal matrix.

Let us take another example of a real matrix and check whether it satisfies the condition of a normal matrix.

For a real matrix, the conjugate transpose will be the same as the transpose of matrix.

Therefore, A^H = A^T =

Now, A^TA =

And AA^T =

Thus, A^TA = AA^T, A is normal.

With this above example we can also say that orthogonal matrices are also normal matrix.

Properties of Normal Matrix

There are a few interesting properties of a Normal matrix are listed below which are useful in deriving many more results in Linear Algebra.

• If A is a normal matrix then it is diagonalizable by a unitary matrix.
• A Hermitian matrix is a normal matrix.

Let A be a Hermitian matrix then A^H = A, then

Now, A^HA − AA^H = AA − AA = A² − A² = 0

⇒ A^HA − AA^H = 0 ⇒ A^HA = AA^H

⇒ A is normal

• A skew-Hermitian matrix is a normal matrix.
• A unitary matrix is normal.

If U is a unitary matrix, then U^HU = UU^H = I, hence normal.

• A symmetric and a skew-symmetric matrix both are normal matrices.
• A normal matrix need not be a Hermitian, skew-Hermitian, Unitary or symmetric matrix.
• An orthogonal matrix is also a normal matrix.
• If A is normal then, AA^H is a Hermitian matrix.
• If A is normal, then there exists a set of orthonormal eigenvectors of A in Cⁿ.
• A normal matrix is unitary if and only if its eigenvalues lies on the unit circle in complex plane.

Related Articles:

• Types of Matrices
• Transpose of Matrix
• Symmetric and Skew-Symmetric Matrix
• Eigenvalue of a Matrix
• Unitary Matrix

Solved Examples on Normal Matrix

Example 1:

Check whether the given matrix

Solution:

The given matrix

We know a normal matrix is unitarily diagonalizable.

Hence, the given matrix is unitarily diagonalizable.

Example 2:

If A =

Solution:

Given, A =

Clearly, A is a skew-symmetric matrix.

Hence, A is a normal matrix.