Hermitian Matrix
Hermitian matrix has a similar property as the symmetric matrix and was named after a mathematician Charles Hermite. The hermitian matrix has complex numbers as its elements, and it is equal to its conjugate transpose matrix.
Let us learn more about hermitian matrix, properties of hermitian matrix, terms related to hermitian matrix, and examples on hermitian matrix.
What Is a Hermitian Matrix?
A hermitian matrix is a square matrix, which is equal to its conjugate transpose matrix. The nondiagonal elements of a hermitian matrix are all complex numbers. The complex numbers in a hermitian matrix are such that the element of the ith row and jth column is the complex conjugate of the element of the jth row and ith column.
The matrix A can be referred to as a hermitian matrix if A = A^{T}. A hermitian matrix is similar to a symmetric matrix but has complex numbers as the elements of its nonprincipal diagonal.
Examples of Hermitian Matrix
Hermitian Matrix of Order 2 x 2: Here the nondiagonal are complex numbers. Only the first element of the first row and the second element of the second row are real numbers. And the complex number of the first row second element is a conjugate complex number of the second row first element.
\(\begin{bmatrix}3& 3 2i \\3 +2 i & 2\end{bmatrix}\)
Hermitian Matrix of Order 3 x 3: Here the nondiagonal elements are all complex numbers. The elements connecting the diagonal from the first row first element to the thirdrow third element are all real numbers. And the other non diagonal elements are conjugate complex numbers of each other.
\(\begin{bmatrix}1 & 2+ i & 5 4i\\2  i & 4 & 6i\\5 + 4i &6i & 2\end{bmatrix}\)
Properties of Hermitian Matrix
The following properties of the hermitian matrix help in a better understanding of a hermitian matrix.
 The elements of the principal diagonal of a hermitian matrix are all real numbers.
 The nondiagonal elements of a hermitian matrix are complex numbers.
 Every hermitian matrix is a normal matrix, such that A^{H} = A.
 The sum of any two hermitian matrices is hermitian.
 The inverse of a hermitian matrix is a hermitian.
 The product of two hermitian matrices is hermitian.
 The determinant of a hermitian matrix is real.
Terms Related to Hermitian Matrix
The following terms are helpful in understanding and learning more about the hermitian matrix.
 Principal Diagonal: In a square matrix, all the set of elements connecting the first element of the first row to the last element of the last row, represents a principal diagonal.
 Symmetric Matrix: A matrix is said to be a symmetric matrix if the transpose of a matrix is equal to the given matrix. A^{T} = A.
 Conjugate Matrix: The conjugate matrix of a given matrix is obtained by replacing the corresponding elements of the given matrix, with its conjugate complex numbers.
 Transpose Matrix: The transpose of a matrix A is represented as A^{T}, and the transpose of a matrix is obtained by changing the rows into columns and columns into rows for a given matrix.
Related Topics
The following topics help in a better understanding of the hermitian matrix.
Examples on Hermitian Matrix

Example 1: Find if the matrix \(\begin{bmatrix}1& 4 + 3i \\4  3 i & 5\end{bmatrix}\) is a hermitian matrix.
Solution:
The given matrix is A = \(\begin{bmatrix}1& 4 + 3i \\4  3 i & 5\end{bmatrix}\).
Conjugate of A = \(\begin{bmatrix}1& 4  3i \\4 + 3 i & 5\end{bmatrix}\)
Transpose of Conjugate of A = \(\begin{bmatrix}1& 4 + 3i \\4  3 i & 5\end{bmatrix}\) = A
Hence the given matrix is a hermitian matrix.

Example 2: Find the sum of these two hermitian matrices, \(\begin{bmatrix}1 & 1+ i & 4 5i\\1  i & 3 & 3i\\4 + 5i &3i & 2\end{bmatrix}\), \(\begin{bmatrix}5 & 1+ i & 3 2i\\1  i & 1 & 2+i\\3 + 2i &2i & 4\end{bmatrix}\), and show that the result is also a hermitian matrix.
Solution:
The given two hermitian matrices are as follows.
A = \(\begin{bmatrix}1 & 1+ i & 4 5i\\1  i & 3 & 3i\\4 + 5i &3i & 2\end{bmatrix}\)
B = \(\begin{bmatrix}5 & 1+ i & 3 2i\\1  i & 1 & 2+i\\3 + 2i &2i & 4\end{bmatrix}\)
A + B = \(\begin{bmatrix}6 & 2+ 2i & 7 7i\\2  2i & 2 & 2+4i\\7 + 7i &24i & 2\end{bmatrix}\)
Conjugate of (A + B) = \(\begin{bmatrix}6 & 2 2i & 7 +7i\\2 + 2i & 2 & 24i\\7  7i &2+4i & 2\end{bmatrix}\)
Transpose of Conjugate of (A + B) = \(\begin{bmatrix}6 & 2+ 2i & 7 7i\\2  2i & 2 & 2+4i\\7 + 7i &24i & 2\end{bmatrix}\) = A + B
Therefore, the sum of two hermitian matrices is also a hermitian matrix.
FAQs on Hermitian Matrix
What Is a Hermitian Matrix?
A hermitian matrix is a square matrix, which is equal to its conjugate transpose matrix. The nondiagonal elements of a hermitian matrix are all complex numbers, and the element of the ith row and jth column is the complex conjugate of the element of the jth row and ith column
How Do You Know If a Matrix Is Hermitian Matrix?
The given matrix can be identified as a hermitian matrix if the complex number of the ith row and jth column is the conjugate complex of the element of the jth row and ith column
What Are the Properties of Hermitian Matrix?
The following properties of a hermitian matrix.
 The elements of the principal diagonal of a hermitian matrix are all real numbers.
 The nondiagonal elements of a hermitian matrix are complex numbers.
 Every hermitian matrix is a normal matrix, such that A^{H} = A.
Is a Hermitian Matrix Also a Symmetric Matrix?
The hermitian matrix is similar to a symmetric matrix because the corresponding elements of the given matrix and the hermitian matrix are conjugate complex numbers.
What Is the Order of a Hermitian Matrix?
A hermitian matrix is a square matrix, with equal number of rows and columns, and has an order n x n.
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