Matrix A = (3 -5 1 -1
4 -1 2 0
0 2 -1 -1
3 0 -3 1)
Express A as the sum of a symmetric matrix U and a skew-symmetric matrix V such that A = U+V
Two unkowns in this Q? How can i solve this?
You are looking for the solution of
$\displaystyle \left [ \begin{matrix} 3 & -5 & 1 & -1 \\ 4 & -1 & 2 & 0 \\0 & 2 & -1 & -1 \\ 3 & 0 & -3 & 1 \end{matrix} \right ] = \left [ \begin{matrix} a & b & c & d \\ b & e & f & g \\c & f & h & i \\ d & g & i & j \end{matrix} \right ] + \left [ \begin{matrix} 0 & k & m & n \\ -k & 0 & p & q \\-m & -p & 0 & r \\ -n & -q & -r & 0 \end{matrix} \right ]$
a, e, h, and j are easy. For the others you have something like:
$\displaystyle -5 = b + k$
and
$\displaystyle 4 = b - k$
Solve for b and k.
It'll take a while, but it can be done.
Or perhaps you will simply want to calculate
$\displaystyle U = \frac{1}{2} \cdot \left ( A + A^T \right )$
and
$\displaystyle V = \frac{1}{2} \cdot \left ( A - A^T \right )$
-Dan