Skew-Hermitian matrix in Discrete mathematics

The skew-symmetric matrix and the skew-Hermitian matrix both have many similarities. In case of the skew-symmetric matrix, the negation of the given matrix and transpose of the matrix are both similar to each other. Similarly, in case of a skew-Hermitian matrix, the negation of the given matrix and the conjugate transpose of the matrix are similar to each other. We can also call the skew-Hermitian matrix as the anti-Hermitian matrix. In this section, we are going to learn about the skew-hermitian matrix, its formula, its properties, its examples, Decomposing matrix into Hermitian and Skew-Hermitian matrices, and many more.

What is Skew-Hermitian matrix

A matrix will be known as the skew-Hermitian matrix if there is a square matrix in which the negation of the matrix and its conjugate transpose matrix are similar to each other, i.e., A^H = -A. Here A^H is used to indicate the conjugate transpose of matrix A. It can also be indicated by the symbol A∗. Now we will learn about A^H to better understand the concept of a skew-Hermitian matrix. We can get the conjugate transpose of matrix A if we replace every element of the transpose of A (i.e., A^T) with its complex conjugate. If there is a complex number x + iy, the complex conjugate of that number will be x - iy. Now we will learn it by an example, which is described as follows:

Example: In this example, we have a matrix A, and we have to show how this matrix is a skew-Hermitian matrix. The elements of matrix A are described as follows:

Skew-Hermitian matrix in Discrete mathematics

Solution: The transpose of matrix A is described as follows:

Now we will do the conjugate transpose of matrix A in the following way:

Now we will do the negation of matrix A in the following way:

From equation 1 and equation 2, we can see that

A^H = -A

Hence matrix A is a skew-Hermitian matrix.

Suppose A is a skew-Hermitian matrix, and A and A^H are used to contain the elements xij and x̅ij, respectively. If these elements are present at the position ith row and jth column, in this case, xij = -x̅ji. This statement shows that a square matrix A will be a skew-hermitian matrix if and only if it contains the following conditions:

A^H = -A OR

xij = -x̅ji

Formula of skew-Hermitian matrix

There are some interesting facts related to the skew-Hermitian matrix with any number of orders. These facts are described as follows:

There will be a purely imaginary value or zero in the diagonal entries of the skew-symmetric matrix.
All the elements of the skew-Hermitian matrix other than the diagonal elements may contain the real and imaginary parts also.
If the ith row and jth column of this matrix except the diagonal elements contain the imaginary part, in this case, both the positions will be the same.
If the ith row and jth column of this matrix except the diagonal elements contain the real part, in this case, both the positions will be the same but with different signs.

The diagonal elements of the above example of a skew-Hermitian matrix are purely imaginary (or they will be zero). In the above example, we can also see that a12 = 1+i and a21 = -1+i. On the basis of these things, it is possible to construct the formula of a skew-Hermitian matrix with an order 2∗2. So in the following form, we can show the 2∗2 skew-Hermitian matrix:

Here x, y, z, and w are used to indicate the real numbers.

Similarly, we can also construct the formula of a skew-Hermitian matrix with an order 3∗3. So in the following form, we can show the 3∗3 skew-Hermitian matrix:

Properties of Skew-Hermitian matrix

There are various properties of a skew hermitian matrix, and some of them are described as follows:

If there is a skew-symmetric matrix and the elements of this matrix are real numbers, then this type of matrix will always be a skew-Hermitian matrix.
In case of a skew-Hermitian matrix, the diagonal elements will always contain either imaginary numbers or zero.
This matrix will be diagonalizable.
In case of a skew-Hermitina matrix, there will either be zeros or purely imaginary numbers in the eigenvalues.
If there is a skew-Hermitian matrix A, then Aⁿ will also be a skew-Hermitian matrix if and only if n is odd. The matrix Aⁿ will be the Hermitian matrix if and only if n is even.
If we perform the addition or subtraction of two skew-Hermitian matrices, then the resultant matrix will be a skew-Hermitian matrix.
If we perform the multiplication of two Hermitian matrices, then the resultant matrix will be a Hermitian matrix.
If we multiply a scalar value and a skew-Hermitian matrix, then the resulting matrix must be a skew-He0rmitian matrix.
If there is a skew-Hermitian matrix A, then iA will be a Hermitian matrix.

Decomposing matrix into Hermitian and Skew-Hermitian matrix

If there is a square matrix, then we can write this matrix as the addition of a Hermitian matrix X and skew-Hermitian matrix Y.

Suppose there is a square matrix A. Then,

A = X + Y

A = (1/2) (A + A^H) + (1/2) (A - A^H). Here A^H is used to indicate the transpose of conjugate matrix A.

Where

X = (1/2) (A + A^H) and

Y = (1/2) (A - A^H)

It shows that

If a hermitian matrix is indicated by A + A^H, then (1/2) × (A + A^H) will also indicate a hermitian matrix.
If a skew-hermitian matrix is indicated by A - A^H, then (1/2) × (A - A^H) will also indicate a skew-hermitian matrix.

Hence if there is a square matrix, then we can write this matrix as the addition of a skew-hermitian matrix and hermitian matrix.

Important points

It is not important that if there is a normal matrix, then it will also be a skew-Hermitian matrix. There are some important points which we should know while learning the concept of a skew-Hermitian matrix. These points are described as follows:

If we do the conjugate of a skew-Hermtitian matrix, then we will always get a skew-Hermitian matrix as a result.
If we do the transpose of a skew-Hermtitian matrix, then we will always get a skew-Hermitian matrix as a result.
If we do the trace of a skew-Hermtitian matrix, then it will be either imaginary or zero.
If there is a square matrix A, then A - A∗ will be a skew-Hermitian matrix.
If we compute the determinant of an odd-order skew-Hermitian matrix, then we will get zero as the result of the determinant.

Examples of skew-Hermitian matrix

There are a lot of examples of Skew-Hermitian matrices, and some of them are described as follows:

Example 1: In this example, we have a matrix A, and we have to determine whether it is a Skew-Hermitian matrix. The elements of matrix A are described as follows:

Solution: First, we will compute transpose of the given matrix A in the following way:

Now we will do the conjugate of the above transpose matrix in the following way:

Now we will do the negation of given matrix A in the following way:

From equation 1 and equation 2, we can see that

A^H = -A

So the given matrix is a skew-Hermitian matrix.

Answer: Matrix A is a skew-Hermitian matrix.

Example 2: In this example, we have two skew-Hermitian matrices, and we have to determine the addition of these two matrices. We have to also prove that this addition's output is also a skew-Hermitian matrix. The elements of these two skew-Hermitian matrices are described as follows:

Solution: From the question, we have two skew-Hermitian matrices, A and B. The addition of these two matrices is described as follows:

Now we will compute transpose of this addition in the following way:

Now we will do the conjugate transpose of the A+B in the following way:

Now we will do the negation of the A+B in the following way:

From equation 1 and equation 2, we can see that

(A+B)^H = - (A+B)

Answer: The addition of two skew-Hermitian matrices will be a skew-Hermitian matrix.

Example 3: In this example, we have a matrix A, and we have to decompose it in the form of addition of Hermitian matrix and skew-Hermitian matrix. The elements of matrix A are described as follows: