Express Matrix as Sum of Symmetric and Skew-Symmetric Matrices
Given:
\[ A = \begin{bmatrix} 4 & 2 & -1 \\ 3 & 5 & 7 \\ 1 & -2 & 1 \end{bmatrix} \]
Formula Used:
\[ \text{Symmetric part } (S) = \frac{1}{2}(A + A^T), \quad \text{Skew-symmetric part } (K) = \frac{1}{2}(A – A^T) \]
Step 1: Find AT
\[ A^T = \begin{bmatrix} 4 & 3 & 1 \\ 2 & 5 & -2 \\ -1 & 7 & 1 \end{bmatrix} \]
Step 2: Find Symmetric Matrix S
\[ A + A^T = \begin{bmatrix} 8 & 5 & 0 \\ 5 & 10 & 5 \\ 0 & 5 & 2 \end{bmatrix} \]
\[ S = \begin{bmatrix} 4 & \tfrac{5}{2} & 0 \\ \tfrac{5}{2} & 5 & \tfrac{5}{2} \\ 0 & \tfrac{5}{2} & 1 \end{bmatrix} \]
Step 3: Find Skew-Symmetric Matrix K
\[ A – A^T = \begin{bmatrix} 0 & -1 & -2 \\ 1 & 0 & 9 \\ 2 & -9 & 0 \end{bmatrix} \]
\[ K = \begin{bmatrix} 0 & -\tfrac{1}{2} & -1 \\ \tfrac{1}{2} & 0 & \tfrac{9}{2} \\ 1 & -\tfrac{9}{2} & 0 \end{bmatrix} \]
Final Result:
\[ A = S + K \]
Thus, the matrix is expressed as the sum of a symmetric and a skew-symmetric matrix.