## Hermitian matrix in Discrete mathematicsThe symmetric matrix and the Hermitian matrix are both used to contain similar properties. The name Hermitian was placed after a famous mathematician Charles Hermite. The Hermitian matrix is used to contain complex numbers in the form of its elements. The ## What is Hermitian matrixA matrix will be known as the Hermitian matrix if there is a square matrix which is similar to its conjugate transpose matrix. In case of a Hermitian matrix, all the A matrix will be known as the Hermitian matrix if it contains the condition, i.e., A = A ## Hermitian matrix of Order 2∗2In case of a Hermitian matrix, the non-diagonal elements will be complex numbers. There will be real numbers in the first element of 1 ## Hermitian matrix of order 3∗3In case of a Hermitian matrix, the non-diagonal elements will be complex numbers. The real numbers will be contained by all those elements which connect the diagonal from first element of 1 The 1 ## Formula of Hermitian matrixWith the help of above described two matrices, we know that the real numbers will be contained by the diagonal elements of the Hermitian matrix. We also know that the elements which exist at the position (i, j) will be the complex conjugate of the element which exists at the position (j, i). Hence the standard form to represent the Hermitian matrix with a 2∗2 is described as follows: Here x, y, z, and w are used to indicate the real numbers. Similarly, the standard form to represent the Hermitian matrix with an order 3∗3 is described as follows: ## Properties of Hermitian matrixThere are various properties of a Hermitian matrix, and some of them are described as follows: - In case of a Hermitian matrix, all the principal diagonal elements will always contain the real numbers.
- There must be complex numbers in the non-diagonal elements.
- If there is a Hermitian matrix, then that matrix must be a normal matrix like A
^{H}= A. - If we perform the addition of two Hermitian matrices, then the resultant matrix will be a Hermitian matrix.
- If we perform the inverse of two Hermitian matrices, then the resultant matrix will be a Hermitian matrix.
- If we perform the multiplication of two Hermitian matrices, then the resultant matrix will be a Hermitian matrix.
- If we calculate the determinant of a Hermitian matrix, then the matrix will be real.
- The eigenvalues are used to contain the real numbers in the Hermitian matrix.
## Terms related to Hermitian matrixThere are some important points which we should know while learning the concept of a Hermitian matrix. These points are described as follows:
## Writing matrix as Hermitian and Skew-HermitianIf there is a square matrix, then we can write this matrix as the addition of a Hermitian matrix P and skew-Hermitian matrix Q. With the help of following formula, we can determine the addition of a skew-Hermitian matrix and Hermitian matrix. Suppose there is a A = (1/2) (A + A - 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. ## Examples of Hermitian matrixThere are a lot of examples of Hermitian matrices, and some of them are described as follows:
Now we will do the conjugate of this matrix like this: Now we will do the transpose of conjugate of A in the following way: Hence the given matrix A is a Hermitian matrix.
det A = xw - (y + zi)(y - zi) = xw - (y = xw - (y = xw - y = a real number In the same way, we can assume a Hermitian matrix with some other order and see that the determinant of that matrix is also a real number. Hence it is proved that the determinant of a Herminant matrix is always a real number.
Now we will do the conjugate of the above addition (A+B) is described as follows: Now we will do the transpose of conjugate of A+B in the following way: It is proved that the addition of two Hermitian matrices will also be a Hermitian matrix.
Now we will do the conjugate of matrix A in the following way: Now we will do the conjugate transpose of A (A∗) in the following way: Here we can see that the transpose conjugate of A and the given matrix A are similar to each other. Hence it is a Hermitian matrix.
The multiplication of xA will be Hermitian if it satisfies the following condition: (xA)∗ = xA Now (xA) = x̅A∗ = x̅A ≠ xA Since x is used to indicate the complex number, which is not similar to its conjugate. Hence (xA)∗ = xA Thus, xA will not be a Hermitian matrix. |