Skew Hermitian Matrix
A skew Hermitian matrix is closely defined just as a skewsymmetric matrix. A skewsymmetric 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.
1.  What is a Skew Hermitian Matrix? 
2.  Skew Hermitian Matrix Formula 
3.  Properties of Skew Hermitian Matrix 
4.  Decomposing Matrix into Hermitian and SkewHermitian 
5.  FAQs on Skew Hermitian Matrix 
What is a Skew Hermitian Matrix?
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 = \(\begin{equation}
\left[\begin{array}{cc}
3 i & 1i \\
1i & i
\end{array}\right]
\end{equation}\) ... (1)
Now,  A = \(\begin{equation}
\left[\begin{array}{cc}
3 i & 1i \\
1i & i
\end{array}\right]
\end{equation}\) ... (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 antiHermitian matrix.
Skew Hermitian Matrix Formula
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 \(\begin{equation}
\left(\begin{array}{cc}
a i & b+c i & c +di\\
b+ci & ei & g+hi \\
c+di & g+hi & ki
\end{array}\right)
\end{equation}\)
Properties of Skew Hermitian Matrix
 If A is a skewsymmetric matrix with all entries to be the real numbers, then it is obviously a skewHermitian 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 A^{n} is also skew Hermitian if n is odd and A^{n} 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.
Decomposing Matrix into Hermitian and SkewHermitian
Any square matrix A can be split into the sum of a Hermitian matrix X and a skewHermitian 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 skewHermitian.
☛ Related Topics:
Skew Hermitian Matrix Examples

Example 1: Is A = \(\begin{equation}
\left[\begin{array}{cc}
5i & 32i \\ \\
32i & i
\end{array}\right]
\end{equation} \) skew Hermitian?Solution:
The transpose of the given matrix is, A^{T} = \(\begin{equation}
\left[\begin{array}{cc}
5i & 32i \\ \\
32i & i
\end{array}\right]
\end{equation} \)Its conjugate is,
A^{H} = \(\begin{equation}
\left[\begin{array}{cc}
5i & 3+2i \\ \\
3+2i & i
\end{array}\right]
\end{equation} \) ... (1)The negative of the given matrix is,
A = \(\begin{equation}
\left[\begin{array}{cc}
5i & 3+2i \\ \\
3+2i & i
\end{array}\right]
\end{equation} \) ... (2)From (1) and (2),
A^{H} = A
Answer: Therefore, the given matrix is skew Hermitian.

Example 2: Prove that the sum of two skew Hermitian matrices \(A = \begin{equation}
\left[\begin{array}{ccc}
0 & 1i & 2 \\
1i & 3 i & i \\
2 & i & 0
\end{array}\right]
\end{equation}\) and \(B = \begin{equation}
\left[\begin{array}{ccc}
i & 2+i & 2 \\
2+i & 2 i & 3+i \\
2 & 3+i & 5i
\end{array}\right]
\end{equation}\) is skew Hermitian.Solution:
The sum of the given two matrices is:
A + B
= \(\begin{equation}
\left[\begin{array}{ccc}
i & 1 & 4 \\
1 & 5 i & 3+2i \\
4 & 3+2i & 5i
\end{array}\right]
\end{equation}\)Now, (A + B)^{T} = \(\begin{equation}
\left[\begin{array}{ccc}
i & 1 & 4 \\
1 & 5 i & 3+2i \\
4 & 3+2i & 5i
\end{array}\right]
\end{equation}\)Its conjugate is, (A + B)^{H} = \(\begin{equation}
\left[\begin{array}{ccc}
i & 1 & 4 \\
1 & 5 i & 32i \\
4 & 32i & 5i
\end{array}\right]
\end{equation}\) ... (1)Now, (A + B) = \(\begin{equation}
\left[\begin{array}{ccc}
i & 1 & 4 \\
1 & 5 i & 32i \\
4 & 32i & 5i
\end{array}\right]
\end{equation}\) ... (2)From (1) and (2),
(A + B)^{H} =  (A + B)
Answer: The sum of two skew Hermitian matrices is skew Hermitian.

Example 3: Decompose the following matrix as the sum of Hermitian and skew Hermitian matrices: A = \(\begin{equation}
\left[\begin{array}{cc}
5i & 12i \\ \\
3 & i
\end{array}\right]
\end{equation} \).Solution:
The given matrix is, A = \(\begin{equation}
\left[\begin{array}{cc}
5i & 12i \\ \\
3 & i
\end{array}\right]
\end{equation} \).Its complex conjugate is:
A^{H} = \(\begin{equation}
\left[\begin{array}{cc}
5i & 3 \\ \\
1+2i & i
\end{array}\right]
\end{equation} \)We know that A = X + Y, where:
 X is Hermitian where X = (1/2) (A + A^{H})
 Y is skew Hermitian where Y = (1/2) (A  A^{H})
X = (1/2) (A + A^{H})
= (1/2) \(\left(
\left[\begin{array}{cc}
5i & 12i \\ \\
3 & i
\end{array}\right]
+ \left[\begin{array}{cc}
5i & 3 \\ \\
1+2i & i
\end{array}\right]
\right)\)= (1/2) \(\left[\begin{array}{cc}
0 & 42i \\ \\
4+2i & 0
\end{array}\right]\)= \(\left[\begin{array}{cc}
0 & 2i \\ \\
2+i & 0
\end{array}\right]\)Y = (1/2) (A  A^{H})
= (1/2) \(\left(
\left[\begin{array}{cc}
5i & 12i \\ \\
3 & i
\end{array}\right]
 \left[\begin{array}{cc}
5i & 3 \\ \\
1+2i & i
\end{array}\right]
\right)\)= (1/2) \(\left[\begin{array}{cc}
10i & 22i \\ \\
22i & 2i
\end{array}\right]\)= \(\left[\begin{array}{cc}
5i & 1i \\ \\
1i & i
\end{array}\right]\)Answer: A = \(\left[\begin{array}{cc}
0 & 2i \\ \\
2+i & 0
\end{array}\right]\) + \(\left[\begin{array}{cc}
5i & 1i \\ \\
1i & i
\end{array}\right]\).
FAQs on Skew Hermitian Matrix
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 skewsymmetric. 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.
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