Hermitian and Anti Hermitian Matrices

1. Hermitian Matrix: Any Complex Square Matrix is called a Hermitian Matrix if the Elements present in it's \(i^{th}\) Row and \(j^{th}\) Column are Complex Conjugates of Elements present in it's \(j^{th}\) Row and \(i^{th}\) Column and has Only Real Values in the Main Diagonal. That is, for any Hermitian Matrix \(A\), \(a_{ij}=\overline{a_{ji}}\) for all \(i\neq j\). Following are some examples of Hermitian Matrices
   
   \(\begin{bmatrix} -2 & 3+2i & 5-4i\\ 3-2i & 1 & 6-2i\\ 5+4i & 6+2i & -5\end{bmatrix}\hspace{.5cm} \begin{bmatrix} 3.5 & -3+7i \\ -3-7i & 0 \end{bmatrix}\)
   
   Please note that for any Hermitian Matrix \(A\), the Conjugate Transpose of the Matrix is the Matrix itself. That is
   
   \(A^\dagger=A\)
2. Anti-Hermitian Matrix: Any Square Matrix is called a Anti-Hermitian Matrix if all its Main Diagona are either 0 or Pure Imaginary Values and the Elements present in it's \(i^{th}\) Row and \(j^{th}\) Column are Negative of Complex Conjugates of the Elements present in it's \(j^{th}\) Row and \(i^{th}\) Column. That is, for any Anti-Hermitian Matrix \(A\), \(a_{ij}=-\overline{a_{ji}}\) for all \(i\neq j\). Following are some examples of Anti-Hermitian Matrices
   
   \(\begin{bmatrix} 7i & -1.5+4i & 4+4i\\ 1.5+4i & 0 & -.3+7i\\ -4+4i & .3+7i & 0\end{bmatrix}\hspace{.5cm} \begin{bmatrix} 4i & 1.23+5i \\ -1.23+5i & 2i \end{bmatrix}\)
   
   Please note that for any Anti-Hermitian Matrix \(A\), the Conjugate Transpose of the Matrix is the negative of the Matrix. That is
   
   \(A^\dagger=-A\)
3. Any Complex Square Matrix \(A\) can be converted to a pair of Hermitian and Anti-Hermitian Matrices by using the following formula
   
   Hermitian Matrix = \({\Large\frac{A + A^\dagger}{2}}\)
   Anti-Hermitian Matrix = \({\Large\frac{A - A^\dagger}{2}}\)
4. Conversely, given any pair of Hermitian and Anti-Hermitian Matrices, the Original Complex Matrix can be obtained by adding the 2 Matrices.