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 the hermitian matrix and its properties along with examples.

A hermitian matrix is a square matrix, which is equal to its conjugate transpose matrix. The non-diagonal elements of a hermitian matrix are all complex numbers. The complex numbers in a hermitian matrix are such that the element of the i^th row and j^th column is the complex conjugate of the element of the j^th row and i^th column.

The matrix A can be referred to as a hermitian matrix if A = A^H. A hermitian matrix is similar to a symmetric matrix but has complex numbers as the elements of its non-principal diagonal.

Hermitian Matrix of Order 2 x 2

Here the non-diagonal are complex numbers. Only the first element of the first row and the second element of the second row are real numbers. Also, 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 also the non-diagonal elements are all complex numbers. The elements connecting the diagonal from the first row first element to the third-row third element are all real numbers. Also, notice that an element in the position (i, j) is the complex conjugate of the element in the position (j, i). For example, in the matrix below, 2 + i is present in the first row and the second column, whereas it's conjugate 2 - i is present in the second row and first column. The same is the case with other complex numbers as well.

\(\begin{bmatrix}1 & 2+ i & 5 -4i\\2 - i & 4 & 6i\\5 + 4i &-6i & 2\end{bmatrix}\)

From the above two matrices, it is clear that the diagonal elements of a Hermitian matrix are always real. Also, the element in the position (i, j) is the complex conjugate of the element in the position (j, i). Hence, a 2 × 2 Hermitian matrix is of the form \(\left[\begin{array}{cc}
x & y+z i \\ \\
y-z i &w
\end{array}\right]\), where x, y, z, and w are real numbers. Similar, we can construct a 3 × 3 Hermitian matrix using the formula \(\left[\begin{array}{ccc}
a & b+c i & c+d i \\
b-c i &e & g+h i \\
c-d i & g-h i &k
\end{array}\right]\).

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 non-diagonal 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.

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 of the diagonal 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 their corresponding conjugates.
• 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 or columns into rows of a given matrix.

A square matrix A can be written as the sum of a Hermitian matrix P and a skew-Hermitian matrix Q where P = (1/2) (A + A^H) and Q = (1/2) (A - A^H). i.e.,

• A = P + Q where
• P = (1/2) (A + A^H) and
• Q = (1/2) (A - A^H)

For any matrix A, one can easily see that (A + A^H) is Hermitian and (A - A^H) is skew-Hermitian.

☛Related Topics:

• Orthogonal Matrix
• Invertible Matrix
• Adjoint of a Matrix

What is the Definition of a Hermitian Matrix?

A hermitian matrix is a square matrix that is equal to the transpose of its conjugate matrix. The diagonal elements of a hermitian matrix are all real numbers, and the element of the (i, j) position is equal to the conjugate of the element in the (j, i) position.

How Do You Know If a Matrix is Hermitian Matrix?

To know whether the given matrix A is Hermitian:

• Find the conjugate matrix of A by replacing every element with its conjugate.
• Then take the transpose of the resultant matrix, which is A^H.
• If A^H = A, then only it is called a Hermitian matrix.

Is a Symmetric Matrix Always Hermitian?

A symmetric matrix with real elements is always Hermitian. This is because the complex conjugate of a real number is itself.

What are the Properties of the Hermitian Matrix?

The following properties of a hermitian matrix.

• The elements of the principal diagonal of a hermitian matrix are all real numbers.
• The non-diagonal elements of a hermitian matrix are complex numbers.
• Every hermitian matrix is a normal matrix, such that A^H = A, where A^H is the transpose of the conjugate matrix of A.

Is a Hermitian Matrix Also a Symmetric Matrix?

No, it doesn't need to be. If all the elements of a Hermitian matrix are real, then it is symmetric as well.

What is the Order of a Hermitian Matrix?

A hermitian matrix is a square matrix, with an equal number of rows and columns, and has an order n x n.

What is the Difference Between a Symmetric and Hermitian Matrix?

A square matrix A is said to be

• symmetric if A^T = A, where A^T is the transpose of matrix A.
• hermitian if A^H = A, where A^H is the transpose of the conjugate matrix of A.