Skew Hermitian Matrix

A skew Hermitian matrix is closely defined just as a skew-symmetric matrix. A skew-symmetric matrix is a matrix whose transpose is equal to the negative of the matrix. In the same way, a skew Hermitian matrix is a matrix whose conjugate transpose is equal to the negative of the matrix.

Let us learn how to identify a skew Hermitian matrix and its properties along with more examples.

A square matrix (with real/complex entries) A is said to be a skew Hermitian matrix if and only if A^H = -A, where A^H is the conjugate transpose of A, and let us see what is A^H. A^H can be obtained by replacing every element of the transpose of A (i.e., A^T) by its complex conjugate (the complex conjugate of a complex number x + iy is x - iy). This is also dented by A^*.

Example: A = \(\begin{equation}
\left[\begin{array}{cc}
3 i & 1+i \\ \\
-1+i & -i
\end{array}\right]
\end{equation} \) is a skew Hermitian matrix. Let us see how.

A^T = Transpose of A = \(\begin{equation}
\left[\begin{array}{cc}
3 i & -1+i \\ \\
1+i & -i
\end{array}\right]
\end{equation}\)

A^H = Conjugate transpose of A = \(\begin{equation}
\left[\begin{array}{cc}
-3 i & -1-i \\
1-i & i
\end{array}\right]
\end{equation}\) ... (1)

Now, - A = \(\begin{equation}
\left[\begin{array}{cc}
-3 i & -1-i \\
1-i & i
\end{array}\right]
\end{equation}\) ... (2)

From (1) and (2), A^H = -A.

Hence, A is a skew Hermitian matrix.

If \(a_{ij}\) and \(\overline{a_{ij}}\) are the elements of A and A^H respectively that are present in the i^th row and j^th column, then \(a_{ij}\) = - \(\overline{a_{ji}}\), if A is a skew Hermitian matrix. i.e., a square matrix A is a skew Hermitian matrix if:

• A^H = -A (or)
• \(a_{ij}\) = - \(\overline{a_{ji}}\)

A skew Hermitian matrix is also called the anti-Hermitian matrix.

In the above example, we can see that the diagonal elements are purely imaginary (or they can be zeros also). Also, we can see that \(a_{12}\) = 1 + i whereas \(a_{21}\) = -1 + i. Based upon these things, we can construct the formula of a 2x2 skew Hermitian matrix. It is of the form \(\begin{equation}
\left[\begin{array}{cc}
x i & y+z i \\ \\
-y+z i & w i
\end{array}\right]
\end{equation}\), here, x, y, z, and w are any real numbers.

In the same way, the formula for 3x3 skew Hermitian matrix is \(\begin{equation}
\left(\begin{array}{cc}
a i & b+c i & c +di\\
-b+ci & ei & g+hi \\
-c+di & -g+hi & ki
\end{array}\right)
\end{equation}\)

• 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 elements of a skew Hermitian matrix are either purely imaginary or zeros.
• A skew Hermitian matrix is diagonalizable.
• Its eigenvalues are either purely imaginary or zeros.
• If A is skew Hermitian, then Aⁿ is also skew Hermitian if n is odd and Aⁿ is Hermitian (i.e., A^H = A) if n is even.
• The sum/difference of two skew Hermitian matrices is always skew Hermitian.
• The scalar multiple of a skew Hermitian matrix is also skew Hermitian.
• If A is skew Hermitian, then iA is Hermitian.

Any square matrix A can be split into the sum of a Hermitian matrix X and a skew-Hermitian matrix Y where X = (1/2) (A + A^H) and Y = (1/2) (A - A^H). i.e.,

• A = X + Y where
• X = (1/2) (A + A^H) and
• Y = (1/2) (A - A^H)

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

☛ Related Topics:

• Properties of Matrices
• Matrix Calculator
• Types of Matrices
• Matrix Multiplication

What is Skew Hermitian Matrix with Example?

A skew Hermitian matrix is a square matrix A if and only if its conjugate transpose is equal to its negative. i.e., A^H = -A, where A^H is the conjugate transpose of A and is obtained by replacing every element in the transpose of A by its conjugate. Example: \(\begin{equation}
\left[\begin{array}{cc}
i & -2+3 i \\
2+3 i & 2i
\end{array}\right]
\end{equation}\).

What is the Difference Between Hermitian Matrix and Skew Hermitian Matrix?

A matrix A is Hermitian if and only if A^H = A and a matrix A is skew Hermitian if and only A^H = -A. To find A^H:

• First, find the transpose of A.
• Then replace every element in its by its complex conjugate.

What are the Diagonal Elements of a Skew Hermitian Matrix?

If A is a square matrix and A^H is its complex conjugate, then A is skew Hermitian if and only if \(a_{ij}\) = - \(\overline{a_{ji}}\). By this condition, we can say that the diagonal elements of a skew Hermitian matrix are either zeros or purely imaginary.

What are the Eigenvalues of a Skew Hermitian Matrix?

A skew Hermitian matrix is a square matrix A that satisfies A^H = -A. Its eigenvalues are either zeros or purely imaginary numbers.

What is the Condition of Skew Hermitian Matrix?

For a square matrix A to be a skew Hermitian matrix, the condition is \(a_{ij}\) = - \(\overline{a_{ji}}\). Thus, a skew Hermitian matrix of order 2x2 looks like \(\begin{equation}
\left[\begin{array}{cc}
x i & y+z i \\ \\
-y+z i & w i
\end{array}\right]
\end{equation}\) and of order 3x3 looks like \(\begin{equation}
\left(\begin{array}{cc}
a i & b+c i & c +di\\
-b+ci & ei & g+hi \\
-c+di & -g+hi & ki
\end{array}\right)
\end{equation}\).

Is a Real Matrix Skew Hermitian?

A real matrix is Skew Hermitian if it is skew-symmetric. i.e., if A is a real matrix and A^T = -A, then A^H = - A obviously.

Is Zero Matrix a Skew Hermitian Matrix?

Yes, of course, a zero matrix is a skew Hermitian matrix because its conjugate transpose and its negative are always equal.