Linear Algebra... symmetrical and skew-symmetrical matrices..?

Question

Hi! I'm having a little trouble with my homework, I just don't understand what they're asking me to do. It says:

a) Find a symmetric matrix "S" and a skew-symmetric matrix "W" such that A=S+W

b) Show that "S" and "W" are uniquely determined by "A".

Can you please help me?

Thank you!

Ami

Answer 1

A = { a_ij} and A^T = { b_ij = a_ji}, thus A - A^T ={c_ij = a_ij - a_ji} = W1, an anti-symmetric matrix, hence   S1 = A + A^T= { a_ij + a_ji}, a symmetric matrix ----> 2 A =  S1 +  W1----> S = S1 / 2 and W = W1 / 2