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 ∈ Cn×n is said to be a normal matrix if AHA = AAH where AH is the conjugate transpose of A.

The matrix A ∈ Cn×n is a Normal Matrix ⇔ AHA = AAH

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.

The matrix A ∈ Cn×n is a Normal Matrix ⇔ D = UAU-1

where D is a diagonal matrix of A.

Examples of Normal Matrix

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

\(\begin{array}{l}\begin{bmatrix}4i & -1+i \\1-i & 4i \\\end{bmatrix}\end{array} \)

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

Let

\(\begin{array}{l}A=\begin{bmatrix}4i & -1+i \\1-i & 4i \\\end{bmatrix}\end{array} \)

Then, AH = (conjugate of A)T =

\(\begin{array}{l}\begin{bmatrix}-4i & -1-i \\1+i & -4i \\\end{bmatrix}^{T}\end{array} \)
=
\(\begin{array}{l}\begin{bmatrix}-4i & 1+i \\ -1-i & -4i \\\end{bmatrix}\end{array} \)

Then AHA =

\(\begin{array}{l}\begin{bmatrix}-4i & 1+i \\ -1-i & -4i \\\end{bmatrix}\begin{bmatrix}4i & -1+i \\1-i & 4i \\\end{bmatrix}\end{array} \)
=
\(\begin{array}{l}\begin{bmatrix} 18 & 8i \\ -8i & 18 \\\end{bmatrix}\end{array} \)

And AAH =

\(\begin{array}{l}\begin{bmatrix}4i & -1+i \\1-i & 4i \\\end{bmatrix}\begin{bmatrix}-4i & 1+i \\ -1-i & -4i \\\end{bmatrix}\end{array} \)
=
\(\begin{array}{l}\begin{bmatrix} 18 & 8i \\ -8i & 18 \\\end{bmatrix}\end{array} \)

Hence AHA = AAH, 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.

\(\begin{array}{l}A=\begin{bmatrix}1 & -1 \\1 & 1 \\\end{bmatrix}\end{array} \)
,

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

Therefore, AH = AT =

\(\begin{array}{l}\begin{bmatrix}1 & 1 \\-1 & 1 \\\end{bmatrix}\end{array} \)

Now, ATA =

\(\begin{array}{l}\begin{bmatrix}1 & 1 \\-1 & 1 \\\end{bmatrix}\begin{bmatrix}1 & -1 \\1 & 1 \\\end{bmatrix}= \begin{bmatrix}2 & 0 \\0 & 2 \\\end{bmatrix}\end{array} \)

And AAT =

\(\begin{array}{l}\begin{bmatrix}1 & -1 \\1 & 1 \\\end{bmatrix}\begin{bmatrix}1 & 1 \\-1 & 1 \\\end{bmatrix} = \begin{bmatrix}2 & 0 \\0 & 2 \\\end{bmatrix}\end{array} \)

Thus, ATA = AAT, 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 AH = A, then

Now, AHA − AAH = AA − AA = A2 − A2 = 0

⇒ AHA − AAH = 0 ⇒ AHA = AAH

⇒ A is normal

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

If U is a unitary matrix, then UHU = UUH = 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, AAH 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.

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Solved Examples on Normal Matrix

Example 1:

Check whether the given matrix

\(\begin{array}{l}\begin{bmatrix}1 & 0 & 7 \\0 &-1 & 0 \\7& 0 & 2 \\\end{bmatrix}\end{array} \)
is unitarily diagonalizable.

Solution:

The given matrix

\(\begin{array}{l}\begin{bmatrix}1 & 0 & 7 \\0 &-1 & 0 \\7& 0 & 2 \\\end{bmatrix}\end{array} \)
is symmetric matrix, and a symmetric matrix is always normal.

We know a normal matrix is unitarily diagonalizable.

Hence, the given matrix is unitarily diagonalizable.

Example 2:

If A =

\(\begin{array}{l}\begin{bmatrix}1 & 2\\-2 &-1 \\\end{bmatrix}\end{array} \)
, then show that A is a normal matrix.

Solution:

Given, A =

\(\begin{array}{l}\begin{bmatrix}1 & 2\\-2 &-1 \\\end{bmatrix}\end{array} \)

Clearly, A is a skew-symmetric matrix.

Hence, A is a normal matrix.

Frequently Asked Questions on Normal Matrix

Q1

How do you know if a given matrix is normal?

A matrix when pre and post multiplied with its conjugate transpose commutes, then the matrix is said to be a normal matrix.

Q2

Is a Hermitian matrix normal?

Yes, a Hermitian matrix is normal.

Q3

Is a normal matrix unitarily diagonalizable?

Yes, a normal matrix is unitarily diagonalizable.

Q4

Is a symmetric matrix normal?

Yes, a symmetric matrix is normal.

Q5

What is condition for a normal matrix to be unitary?

A normal matrix is unitary if and only if its eigenvalues lies on the unit circle in complex plane

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