Proving Unique Decomposition of a Square Matrix

[mlarson9000]

Homework Statement

An nxn matrix C is skew symmetric if C^t = -C. Prove that every square matrix A can be written uniquely as A = B + C where B is symmetric and C is skew symmetric.

Homework Equations

The Attempt at a Solution

No clue.

 

[Dick]

You aren't trying very hard, now are you? Tell me about (A+A^T) and (A-A^T). Are they symmetric, skew-symmetric or neither? You have to help here.

 

[mlarson9000]

(A+A^T)is symmetric, (A-A^T)is skew symmetric, but adding them together produces 2A, not A. I'm not sure what to do with this information.

 

[Dick]

(A+A^T)is symmetric, (A-A^T)is skew symmetric, but adding them together produces 2A, not A. I'm not sure what to do with this information.

That's not such a big problem. Divide each one by two.

 

[mlarson9000]

That's not such a big problem. Divide each one by two.

How embarrassing.