Question

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

Hint:

Use P=(B+B^T)/2 for the symmetric part and Q=(B-B^T)/2 for the skew-symmetric part.

Answer 1

Step 1: Key Formula:
Any square matrix \(B\) can be written as \(B=P+Q\) where \(P=\dfrac12(B+B^{T})\) (symmetric) and \(Q=\dfrac12(B-B^{T})\) (skew-symmetric).

Step 2: Computing B transpose:
\(B^{T}=\begin{bmatrix}2&-1&1\\-2&3&-2\\-4&4&-3\end{bmatrix}\).

Step 3: Computing the symmetric part P:
\(P=\dfrac12(B+B^{T})=\dfrac12\begin{bmatrix}4&-3&-3\\-3&6&2\\-3&2&-6\end{bmatrix}=\begin{bmatrix}2&-3/2&-3/2\\-3/2&3&1\\-3/2&1&-3\end{bmatrix}\).

Step 4: Computing the skew-symmetric part Q:
\(Q=\dfrac12(B-B^{T})=\dfrac12\begin{bmatrix}0&-1&-5\\1&0&6\\5&-6&0\end{bmatrix}=\begin{bmatrix}0&-1/2&-5/2\\1/2&0&3\\5/2&-3&0\end{bmatrix}\).

Final Answer:
\(B=P+Q\) with \(P=\begin{bmatrix}2&-3/2&-3/2\\-3/2&3&1\\-3/2&1&-3\end{bmatrix}\) (symmetric) and \(Q=\begin{bmatrix}0&-1/2&-5/2\\1/2&0&3\\5/2&-3&0\end{bmatrix}\) (skew-symmetric).\[ \boxed{B=P+Q} \]