Symmetric and Skew Symmetric Matrices

1. Symmetric Matrix: Any Square Matrix is called a Symmetric Matrix if the Elements present in it's \(i^{th}\) Row and \(j^{th}\) Column are Same as Elements present in it's \(j^{th}\) Row and \(i^{th}\) Column. That is, for any Symmetric Matrix \(A\), \(a_{ij}=a_{ji}\) for all \(i\neq j\). Following are some examples of Symmetric Matrices
   
   \(\begin{bmatrix} -2 & 3 & 5\\ 3 & 0 & 6\\ 5 & 6 & -5\end{bmatrix}\hspace{.5cm} \begin{bmatrix} 3.5 & -3 \\ -3 & 0 \end{bmatrix}\)
   
   Please note that for any Symmetric Matrix \(A\), the Transpose of the Matrix is the Matrix itself. That is
   
   \(A^T=A\)
2. Skew-Symmetric Matrix: Any Square Matrix is called a Skew-Symmetric Matrix if all its Main Diagonal Elements are 0 and the Elements present in it's \(i^{th}\) Row and \(j^{th}\) Column are Negative of the Elements present in it's \(j^{th}\) Row and \(i^{th}\) Column. That is, for any Skew-Symmetric Matrix \(A\), \(a_{ij}=-a_{ji}\) for all \(i\neq j\). Following are some examples of Skew-Symmetric Matrices
   
   \(\begin{bmatrix} 0 & -1.5 & 4\\ 1.5 & 0 & -.3\\ -4 & .3 & 0\end{bmatrix}\hspace{.5cm} \begin{bmatrix} 0 & 1.23 \\ -1.23 & 0 \end{bmatrix}\)
   
   Please note that for any Skew-Symmetric Matrix \(A\), the Transpose of the Matrix is the negative of the Matrix. That is
   
   \(A^T=-A\)
3. Any Square Matrix \(A\) can be converted to a pair of Symmetric and Skew-Symmetric Matrices by using the following formula
   
   Symmetric Matrix = \({\Large\frac{A + A^T}{2}}\)
   Skew-Symmetric Matrix = \({\Large\frac{A - A^T}{2}}\)
4. Conversely, given any pair of Symmetric and Skew-Symmetric Matrices, the Original Matrix can be obtained by adding the 2 Matrices.