Skew-symmetric matrices and subspaces

[Catchfire]

Homework Statement

Let W1 be the set of all nxn skew-symmetric matrices with entries from a field F. Assume F is not characteristic 2 and let W2 be a subspace of M_nxn(F) consisting of all nxn symmetric matrices. Prove the direct sum of W1 and W2 is M_nxn(F).

Homework Equations

The Attempt at a Solution

I'll just do this for n = 3 for ease of formatting.

Assume A is in W1 and B is in W2.
A = $\left( \begin{array}{ccc}<br /> 0 & b & c \\<br /> -b & 0 & d \\<br /> -c & -d & 0 \end{array} \right)$

B = $\left( \begin{array}{ccc}<br /> e & f & g \\<br /> f & e & h \\<br /> g & h & e \end{array} \right)$

A+B = $\left( \begin{array}{ccc}<br /> e & b+f & c+g \\<br /> f-b & e & d+h \\<br /> g-c & h-d & e \end{array} \right)$

For elements a_x,y = a_y,x in A they must both equal 0. ∴ W1 $\cap$ W2 = {0}

So I have one part of the direct sum proof but the W1+W2 = M_nxn(F) part isn't working for me, clearly the diagonal entries can't be all the same.

I guess the first thing I'm wondering about is how to use the assumption that F isn't characteristic 2. I don't see how that assumption helps me at all.

 

[Dick]

The usual way to split a matrix A into symmetric and skew-symmetric is that (A+A^T)/2 is symmetric and (A-A^T)/2 is skew-symmetric (where ^T means transpose) and their sum is A. You are going to hit a glitch in characteristic 2. Because then 2=1+1 is not invertible.

 

[Catchfire]

Interesting I haven't seen those definitions before.
OK so the characteristic 2 condition is more of an edge case issue than anything?

 

[Dick]

Interesting I haven't seen those definitions before.
OK so the characteristic 2 condition is more of an edge case issue than anything?

It's not really a definition. It's a trick to define the projections onto the subspaces. And you could think of characteristic 2 as an edge case, I suppose. But the fundamental problem is that, e.g., the nonzero matrix [[1,1],[1,1]] is both symmetric AND skew-symmetric in characteristic 2. In fact, there is no difference between being symmetric and skew-symmetric. Because x=(-x).

 

[larrypw]

thank you so much dick! you enlightened me about my homework due tmr!