A Hermitian matrix is a special type of square matrix where the matrix is equal to its conjugate transpose. This means if you take the complex conjugate of every element and then swap rows with columns, the matrix stays the same.

Hermitian matrices may have complex numbers, but the elements on the main diagonal are always real numbers (no imaginary part).

A matrix is a group of numbers or symbols arranged neatly in rows and columns. Each item in the matrix is called an element. The order of a matrix tells us how many rows and columns it has. For example, a 2×2 matrix has 2 rows and 2 columns and contains 4 elements in total.

In this topic, we will learn about Hermitian matrices, their properties, and also study skew-Hermitian matrices along with examples for better understanding.

What is Hermitian Matrix?

A Hermitian matrix is a square matrix that stays the same when you take its conjugate transpose. That means if you change each element to its complex conjugate and then switch rows and columns, the matrix does not change.

Hermitian matrices can have complex numbers (numbers that include both real and imaginary parts), but all the numbers on the main diagonal are always real numbers.

A complex number is written in the form a + ib, where:

• a is the real part, and
• b is the imaginary part (with i being the square root of -1).

The name hermitian comes from a French Mathematician Charles Hermite (1822 – 1901). Mathematically, a Hermitian matrix is defined as,

A complex square matrix \( A = [a_{ij}]_{nxn}\) such that \( A^{\ast}=A \), where \( A^{\ast} \) is the conjugate transpose of a matrix A. In other words, if ∀ \( a_{ij}\in A \)

\(a_{ij}=a_{ji}^{\ast}\) where \(a_{ji}^{\ast}\) is the complex conjugate of \(a_{ji}\), then A is a hermitian matrix. Remember that the complex conjugate of a complex number a+ib is obtained by changing the sign of its complex part; a-ib.

For example, \( A=\left[_{6-i\ \ \ \ \ \ -1}^{\ \ 8\ \ \ \ \ \ \ \ 6+i}\right] \)

Then the conjugate of A is, \( \overline{A}=\left[_{6+i\ \ \ \ \ \ -1}^{\ \ 8\ \ \ \ \ \ \ \ 6-i}\right] \)

and the transpose conjugate of A is, \( A^{\ast}=\left[_{6-i\ \ \ \ \ \ -1}^{\ \ 8\ \ \ \ \ \ \ \ 6+i}\right] \)

As, \( A^{\ast}=A \),

Therefore A is a hermitian matrix.

Properties of Hermitian Matrix

Some properties of a hermitian matrix are given below:

• The elements of the principal diagonal of a hermitian matrix are always real numbers.
• The non-diagonal elements of the hermitian matrix can be complex numbers.
• If A is a Hermitian matrix, and k is any real scalar, then kA is also a Hermitian matrix. This is because \(\left(kA\right)^{\ast}=kA^{\ast}=kA \), if k is a real number
• The sum of two hermitian matrices is again a hermitian matrix.
• The product of two hermitian matrices is hermitian.
• If A and B are square matrices, then \( \left(AB\right)^{\ast}=B^{\ast}A^{\ast}\). If A and B are Hermitian, then \( \left(AB\right)^{\ast}=BA \).
• The inverse of a hermitian matrix is a hermitian matrix.
• The determinant of a hermitian matrix is always real.
• The conjugate of a hermitian matrix is also a hermitian matrix.
• If A is a hermitian matrix, then \( A^{\ast}A \) and \( AA^{\ast} \) is also hermitian matrix.
• Any square matrix can be represented as A + iB, where A and B are hermitian matrices.
• Given a square matrix A, if \( A^{\ast}=-A \), then A is called the skew-hermitian matrix.
• If A is a Hermitian matrix, then \( A^n \) is also hermitian for all positive integers n.
• The trace of a Hermitian matrix is always real.

Terms Related to Hermitian Matrix 

To better understand Hermitian matrices, it helps to first know a few related terms:

• Principal Diagonal:
  In a square matrix (same number of rows and columns), the principal diagonal is made up of the elements that go from the top-left corner to the bottom-right corner.
• Symmetric Matrix:
  A matrix is called symmetric if it looks the same even when you flip it over its diagonal. In simple terms, if you take the transpose of the matrix and it remains unchanged (Aᵀ = A), it is symmetric.
• Conjugate Matrix:
  To get the conjugate of a matrix, change each number in the matrix to its complex conjugate. That means, if a number has an imaginary part (like 2 + 3i), replace it with its conjugate (2 - 3i).
• Transpose Matrix:
  The transpose of a matrix is made by switching its rows and columns. That means the first row becomes the first column, the second row becomes the second column, and so on. It’s written as Aᵀ.

What is Skew-Hermitian Matrix?

A square matrix is called a skew-hermitian matrix if its conjugate transpose is the negative of the original matrix i.e. \( A^{\ast}=-A \). In other words, if ∀ \( a_{ij}\in A \)
\(a_{ij}=-a_{ji}^{\ast}\) where \(a_{ji}^{\ast}\) is the complex conjugate of \(a_{ji}\), then A is a skew-hermitian matrix.

For example, \( A=\begin{bmatrix}\ \ \ \ \ 0&\ 1+2i&\ \ 3+i\\
-1+2i&\ \ \ \ 3i&\ \ 4+3i\\
-3+i&-4+3i&\ \ \ -2i\end{bmatrix} \)
So, conjugate of A is, \( \overline{A}=\begin{bmatrix}\ \ \ \ \ 0&\ 1-2i&\ \ 3-i\\
-1-2i&\ \ \ \ -3i&\ \ 4-3i\\
-3-i&-4-3i&\ \ \ 2i\end{bmatrix} \)
Now, conjugate transpose of A is, \( A^{\ast}=\begin{bmatrix}\ \ \ \ 0&-1-2i&\ \ -3-i\\
1-2i&\ \ \ \ -3i&\ \ -4-3i\\
3-i&\ \ 4-3i&\ \ \ \ \ \ \ \ 2i\end{bmatrix} \)
As we get, \( A^{\ast}=-A \), therefore the given matrix A is a skew-hermitian matrix.

Any square matrix can be uniquely represented as a sum of a hermitian and a skew-hermitian matrix.
Let us take a matrix, A then
\( A=\frac{1}{2}\left(A+A^{\ast }\right)+\frac{1}{2}\left(A-A^{\ast }\right) \) , where \( \left(A+A^{\ast}\right) \) is hermitian and \(\left(A-A^{\ast }\right) \) is skew-hermitian.

Properties of Skew-Hermitian Matrix

Some properties of a skew-hermitian matrix are given below:

• If A is a skew-symmetric matrix with all entries to be the real numbers, then it is obviously a skew-hermitian matrix.
• The diagonal matrix elements of a skew-hermitian matrix are either complex numbers or zeros.
• A skew hermitian matrix is diagonalizable, which means it can have a lower and upper triangular value zero.
• Its eigenvalues are either purely imaginary or zeros.
• If A is skew-hermitian, then \( A^n \) is also skew-hermitian given n is odd and \( A^n \) is hermitian given n is even.
• The sum or difference of two skew-hermitian matrices is always a skew-hermitian matrix.
• The scalar multiple of a skew-hermitian matrix is also a skew-hermitian matrix.
• If A is skew-hermitian, then iA is Hermitian.

Eigenvalues of a Hermitian Matrix

Eigenvalues of a Hermitian matrix are always real.

Let us consider A to be a hermitian matrix, such that \( A^{\ast}=A \) and \( \lambda \) be the eigenvalue of A, where \( \lambda\ne0 \), such that
\( A\vec{v}=λ\vec{v} \), where \( \vec{v} \) is a non-zero vector.

\( \Rightarrow\left(A\vec{v}\right)^{\ast}=\left(λ\vec{v}\right)^{\ast} \)
\( \Rightarrow\left(\vec{v}^{\ast}A^{\ast}\right)=\left(λ^{\ast}\vec{v}^{ \ast}\right) \)
Multiplying both sides by \( \Rightarrow\vec{v} \), using matrix multiplication we get
\( \Rightarrow\left(\vec{v}^{\ast}A^{\ast}\vec{v}\right)=\left(λ^{\ast}\vec{v}^{\ast}\vec{v}\right) \)

As A is a hermitian matrix so we know that, \( A^{\ast}=A \)

\( \Rightarrow\left(\vec{v}^{\ast}A\vec{v}\right)=\left(λ^{\ast}\vec{v}^{\ast}\vec{v}\right) \)
\( \Rightarrow\left(\vec{v}^{\ast}\lambda\vec{v}\right)=\left(λ^{\ast}\vec{v}^{\ast}\vec{v}\right) \)
\( \Rightarrow\left(\lambda\vec{v}^{\ast}\vec{v}\right)=\left(λ^{\ast}\vec{v}^{\ast}\vec{v}\right) \)
Since \( \vec{v} \) is non-zero, so we can say, \( \vec{v}^{\ast}\vec{v}\ne0 \).

\( \Rightarrow \lambda =\lambda ^{\ast } \)
\( \Rightarrow\lambda\in R \)
Thus the eigenvalues of a hermitian matrix are always real.

Example of Eigenvalues of a Hermitian Matrix: Let us consider, \( A=\left[_{1+i\ \ \ \ \ \ 2}^{\ \ 3\ \ \ \ \ \ \ \ 1-i}\right] \)

This is a hermitian matrix, as \( A^{\ast}=A \)

The characteristic polynomial of A is,

\( \left|A-\lambda I\right|=\left|_{1+i\ \ \ \ \ \ 2-\lambda}^{3-\lambda\ \ \ \ \ 1-i}\right| \)
\( =\left(3-\lambda\right)\left(2-\lambda\right)-\left[\left(1-i\right)\left(1+i\right)\right] \)
\( =\lambda^2-5\lambda+4 \)
\( =\left(\lambda-1\right)\left(\lambda-4\right) \)

Thus, the eigenvalues of A are 1 and 4 which are real.

Solved Examples on Hermitian Matrix

Some solved examples on Hermitian Matrix are given below:

Example 1: Check whether the given matrix is hermitian or not. \( \begin{bmatrix}\ \ \ 1&1+i&\ 4-5i\\ 1-i&\ \ \ 3&\ \ \ 3i\\ 4+5i&-3i&\ -2\end{bmatrix} \)

Solution: Let the given matrix be A.

To check whether the given matrix is hermitian or not, first we have to find the conjugate transpose.
Therefore, conjugate of the given matrix, \( \overline{A\ }=\begin{bmatrix}\ \ \ 1&1-i&\ 4+5i\\
1+i&\ \ \ 3&\ \ -3i\\
4-5i&\ \ 3i&\ \ -2\end{bmatrix} \)
Now we have to transpose it, so conjugate transpose of A is \( A^{\ast}=\begin{bmatrix}\ \ \ 1&\ 1+i&\ 4-5i\\
1-i&\ \ \ \ 3&\ \ \ \ 3i\\
4+5i&\ -3i&\ \ -2\end{bmatrix} \)
Thus we see that the given matrix satisfies the condition \( A^{\ast}=A \) , hence the given matrix is a hermitian matrix.

Example 2: Check whether the matrix
A =
[ 6 − 7i  0 ]
[−99   6 + 7i ]
is a Hermitian matrix.

Solution:
To check if the matrix is Hermitian, we need to find the conjugate transpose of A and compare it with A.

Step 1: Take the complex conjugate of each element in A
Conjugate of A =
[ 6 + 7i  0 ]
[−99   6 − 7i ]

Step 2: Take the transpose of the conjugate (interchange rows and columns):
Conjugate transpose A* =
[ 6 + 7i  −99 ]
[ 0    6 − 7i ]

Step 3: Compare A with the original matrix A*
Original A =
[ 6 − 7i  0 ]
[−99   6 + 7i ]

Since A* is not equal to A, the matrix is not Hermitian.
The given matrix is not a Hermitian matrix.

Example 3: Check whether the matrix
A =
[ 2    3 + i ]
[ 3 − i  5   ]
is a Hermitian matrix.

Solution:
To check if A is Hermitian, we find the conjugate transpose of A and compare it with A.

Step 1: Take the complex conjugate of each element in A
Conjugate of A =
[ 2    3 − i ]
[ 3 + i  5   ]

Step 2: Take the transpose of the conjugate (swap rows and columns):
Conjugate transpose A* =
[ 2    3 + i ]
[ 3 − i  5   ]

Step 3: Compare A with the original matrix A*
Original A =
[ 2    3 + i ]
[ 3 − i  5   ]

Since A* is equal to A, the matrix is Hermitian.

The given matrix is a Hermitian matrix.

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