Question

Show that every n×n matrix A can be written uniquely as a sum A = B + C where B is symmetric (i.e. - B^T=B) and C is skew-symmetric (i.e. - C^T=-C).

Answer 1

The problems before this one asked to find the general form of the symmetric and the skew-symmetric matrices, so that may be of help. I got:\[b_{ij}=b_{ji}\]\[c_{ij}=-c_{ji}\]

Answer 2

That is correct. Go from there.

Answer 3

Obviously one could say from A+B=C the following:\[a_{ij}=b_{ij}+c_{ij}\]\[a_{ij}=b_{ji}-c_{ji}\]I don't understand where to go from there to show that A can be written as a unique sum of B and C.

Answer 4

*A=B+C

Answer 5

That is a system of linear equations.. You can solve it for b_ij and c_ij.

Answer 6

Aahhhh smart. :P

Answer 7

\[ \left\{\begin{array}{l}a_{ij} = b_{ij}+c_{ij}\\a_{ji}=b_{ij}-c_{ij} \end{array}\right. \]
You are given a_ij and a_ji. Now you need to solve this system for b_ij and c_ij!

Answer 8

Thank you :)