A matrix is categorized as a
Normal matrix
if it satisfies a specific condition. This condition states that the matrix, when pre and post multiplied with its conjugate transpose, should commute. In simpler terms, a normal matrix is a matrix that commutes with its own conjugate transpose during
matrix multiplication
.
The concept of a Normal Matrix is a broader classification that includes Unitary, Hermitian, skew-Hermitian matrices, and symmetric and skew-symmetric matrices.
Interestingly, a normal matrix is also unitarily diagonalizable. This means that to diagonalize a normal matrix, we need a unitary matrix as the modal matrix. Let's delve deeper into the concept of a normal matrix.
Defining a Normal Matrix
When we talk about a square matrix A of order n×n consisting of complex numbers (A ∈ C
n×n), it is referred to as a normal matrix if A
H
A = AA
H
where A
H
represents the conjugate transpose of A.
A ∈ C
n×n
is a Normal Matrix ⇔ A
H
A = AA
H
We also refer to the matrix A as a Normal matrix if there is a Unitary matrix U that acts as a modal matrix, which diagonalizes A.
A ∈ C
n×n
is a Normal Matrix ⇔ D = UAU
-1
Here, D is a diagonal matrix of A.
Illustrating with Examples
Let's consider a complex matrix of order 2 × 2,
To ascertain if this matrix is normal, we need to find the conjugate transpose of the given matrix.
Let's assume
Then, A
H
= (conjugate of A)
T
=
=
Then A
H
A =
=
And AA
H
=
=
Hence A
H
A = AA
H
, which is a normal matrix.
Let's consider a real matrix and check if it meets the normal matrix condition.
,
For a real matrix, the conjugate transpose is the same as the transpose of the matrix.
Therefore, A
H
= A
T
=
Now, A
T
A =
And AA
T
=
Thus, A
T
A = AA
T
, A is normal.
From the above example, we can infer that orthogonal matrices are also normal matrices.
Distinguishing Features of a Normal Matrix
A Normal matrix exhibits several interesting properties that are beneficial in deriving numerous results in Linear Algebra.
If A is a normal matrix, it can be diagonalized by a unitary matrix.
A Hermitian matrix qualifies as a normal matrix.
Assuming A to be a Hermitian matrix, if A
H
= A, then
Now, A
H
A − AA
H
= AA − AA = A
2
− A
2
= 0
⇒ A
H
A − AA
H
= 0 ⇒ A
H
A = 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
H
U = UU
H
= I, hence normal.
Both a symmetric and a skew-symmetric matrix are normal matrices.
A normal matrix doesn't necessarily have to be a Hermitian, skew-Hermitian, Unitary, or symmetric matrix.
An orthogonal matrix also fits the definition of a normal matrix.
If A is normal, then AA
H
is a Hermitian matrix.
If A is normal, there exists a set of orthonormal eigenvectors of A in C
n
.
A normal matrix is unitary if and only if its eigenvalues are situated on the unit circle in the complex plane.