Question

Express the matrix
\[ \begin{bmatrix} 7 & 0 & 3 \\ 2 & 4 & 1 \\ -5 & 6 & 8 \end{bmatrix} \] as the sum of symmetric and skew-symmetric matrices.

Hint:

To express a matrix as the sum of a symmetric and a skew-symmetric matrix, use the formulas \( S = \frac{1}{2}(A + A^T) \) and \( K = \frac{1}{2}(A - A^T) \).

Answer 1

Step 1: Definition of symmetric and skew-symmetric matrices.

A matrix \( M \) is symmetric if \( M = M^T \) and skew-symmetric if \( M = -M^T \).

Step 2: Decompose the matrix.

Let \( A \) be the given matrix:
\[ A = \begin{bmatrix} 7 & 0 & 3 \\ 2 & 4 & 1 \\ -5 & 6 & 8 \end{bmatrix} \]
We decompose \( A \) into the sum of a symmetric matrix \( S \) and a skew-symmetric matrix \( K \):
\[ A = S + K \]
where:
\[ S = \frac{1}{2}(A + A^T), \quad K = \frac{1}{2}(A - A^T) \]
Step 3: Calculate \( S \) and \( K \).

First, find the transpose of \( A \):
\[ A^T = \begin{bmatrix} 7 & 2 & -5 \\ 0 & 4 & 6 \\ 3 & 1 & 8 \end{bmatrix} \]
Now, calculate \( S \):
\[ S = \frac{1}{2}\left( \begin{bmatrix} 7 & 0 & 3 \\ 2 & 4 & 1 \\ -5 & 6 & 8 \end{bmatrix} + \begin{bmatrix} 7 & 2 & -5 \\ 0 & 4 & 6 \\ 3 & 1 & 8 \end{bmatrix} \right) \]
\[ S = \frac{1}{2} \begin{bmatrix} 14 & 2 & -2 \\ 2 & 8 & 7 \\ -2 & 7 & 16 \end{bmatrix} = \begin{bmatrix} 7 & 1 & -1 \\ 1 & 4 & \frac{7}{2} \\ -1 & \frac{7}{2} & 8 \end{bmatrix} \]
Next, calculate \( K \):
\[ K = \frac{1}{2}\left( \begin{bmatrix} 7 & 0 & 3 \\ 2 & 4 & 1 \\ -5 & 6 & 8 \end{bmatrix} - \begin{bmatrix} 7 & 2 & -5 \\ 0 & 4 & 6 \\ 3 & 1 & 8 \end{bmatrix} \right) \]
\[ K = \frac{1}{2} \begin{bmatrix} 0 & -2 & 8 \\ 2 & 0 & -5 \\ -8 & 5 & 0 \end{bmatrix} = \begin{bmatrix} 0 & -1 & 4 \\ 1 & 0 & -\frac{5}{2} \\ -4 & \frac{5}{2} & 0 \end{bmatrix} \]
Step 4: Conclusion.

Thus, the matrix \( A \) can be expressed as the sum of a symmetric matrix \( S \) and a skew-symmetric matrix \( K \):
\[ A = \begin{bmatrix} 7 & 1 & -1 \\ 1 & 4 & \frac{7}{2} \\ -1 & \frac{7}{2} & 8 \end{bmatrix} + \begin{bmatrix} 0 & -1 & 4 \\ 1 & 0 & -\frac{5}{2} \\ -4 & \frac{5}{2} & 0 \end{bmatrix} \]