Question

Express the matrix \( A = \begin{bmatrix} 3 & 3 & -1 \\ -2 & -2 & 1 \\ -4 & -5 & 2 \end{bmatrix} \) as the sum of a symmetric and a skew-symmetric matrix.

Hint:

A quick check for your result: P should be symmetric (\( P = P^T \)) and Q should be skew-symmetric (\( Q = -Q^T \)). Also, the diagonal elements of any skew-symmetric matrix must be zero.

Answer 1

Step 1: Understanding the Concept:
Any square matrix A can be uniquely expressed as the sum of a symmetric matrix P and a skew-symmetric matrix Q.
Step 2: Key Formula or Approach:
The symmetric part is given by \( P = \frac{1}{2}(A + A^T) \).
The skew-symmetric part is given by \( Q = \frac{1}{2}(A - A^T) \).
Then, \( A = P + Q \).
Step 3: Detailed Explanation or Calculation:
The given matrix is \( A = \begin{bmatrix} 3 & 3 & -1 \\ -2 & -2 & 1 \\ -4 & -5 & 2 \end{bmatrix} \).
First, find the transpose of A:
\[ A^T = \begin{bmatrix} 3 & -2 & -4 \\ 3 & -2 & -5 \\ -1 & 1 & 2 \end{bmatrix} \]
Calculate the symmetric matrix P:
\[ A + A^T = \begin{bmatrix} 3 & 3 & -1 \\ -2 & -2 & 1 \\ -4 & -5 & 2 \end{bmatrix} + \begin{bmatrix} 3 & -2 & -4 \\ 3 & -2 & -5 \\ -1 & 1 & 2 \end{bmatrix} = \begin{bmatrix} 6 & 1 & -5 \\ 1 & -4 & -4 \\ -5 & -4 & 4 \end{bmatrix} \]
\[ P = \frac{1}{2}(A + A^T) = \frac{1}{2}\begin{bmatrix} 6 & 1 & -5 \\ 1 & -4 & -4 \\ -5 & -4 & 4 \end{bmatrix} = \begin{bmatrix} 3 & 1/2 & -5/2 \\ 1/2 & -2 & -2 \\ -5/2 & -2 & 2 \end{bmatrix} \]
Calculate the skew-symmetric matrix Q:
\[ A - A^T = \begin{bmatrix} 3 & 3 & -1 \\ -2 & -2 & 1 \\ -4 & -5 & 2 \end{bmatrix} - \begin{bmatrix} 3 & -2 & -4 \\ 3 & -2 & -5 \\ -1 & 1 & 2 \end{bmatrix} = \begin{bmatrix} 0 & 5 & 3 \\ -5 & 0 & 6 \\ -3 & -6 & 0 \end{bmatrix} \]
\[ Q = \frac{1}{2}(A - A^T) = \frac{1}{2}\begin{bmatrix} 0 & 5 & 3 \\ -5 & 0 & 6 \\ -3 & -6 & 0 \end{bmatrix} = \begin{bmatrix} 0 & 5/2 & 3/2 \\ -5/2 & 0 & 3 \\ -3/2 & -3 & 0 \end{bmatrix} \]
Step 4: Final Answer:
The matrix A can be expressed as the sum \( A = P + Q \), where:
\[ A = \begin{bmatrix} 3 & 1/2 & -5/2 \\ 1/2 & -2 & -2 \\ -5/2 & -2 & 2 \end{bmatrix} + \begin{bmatrix} 0 & 5/2 & 3/2 \\ -5/2 & 0 & 3 \\ -3/2 & -3 & 0 \end{bmatrix} \]