Question

Express

\[ \begin{bmatrix} 5 \\ 3 \end{bmatrix} \]

as the sum of a symmetric matrix and a skew-symmetric matrix.

Hint:

Any matrix can be expressed as the sum of a symmetric matrix and a skew-symmetric matrix using the formulas \( S = \frac{1}{2} (M + M^T) \) and \( K = \frac{1}{2} (M - M^T) \).

Answer 1

Step 1: Symmetric and skew-symmetric matrices.

A matrix \( M \) can be written as the sum of a symmetric matrix and a skew-symmetric matrix:

\[ M = S + K \]

where \( S \) is symmetric and \( K \) is skew-symmetric.

The symmetric part \( S \) is given by:

\[ S = \frac{1}{2}(M + M^{T}) \]

The skew-symmetric part \( K \) is given by:

\[ K = \frac{1}{2}(M - M^{T}) \]

Step 2: Identifying the given matrix.

Let \[ M = \begin{bmatrix} 5 \\ 3 \end{bmatrix} \] We attempt to express \( M \) as the sum of a symmetric matrix and a skew-symmetric matrix using the above definitions.