Thread: Matrices represented by Symmetric/Skew Symmetric Matrices

I need to show that ever square matrix A can be expressed as the sum of a symmetric matrix and a skew symmetric matrix. I'm completely confused on this question... Should I answer this by filling in an actual matrix with variables, and then somehow come up with the appropriate symmetric matrices, or what?

Originally Posted by Hellreaver

I need to show that ever square matrix A can be expressed as the sum of a symmetric matrix and a skew symmetric matrix. I'm completely confused on this question... Should I answer this by filling in an actual matrix with variables, and then somehow come up with the appropriate symmetric matrices, or what?

It will help you (very much) to write: , where is the transpose matrix of .

But what do I need to do with that? I'm so confused... Sorry for my ignorance on the subject... I just don't get some of these things at all...

Originally Posted by Hellreaver

But what do I need to do with that? I'm so confused..

You asked to write a matrix as a sum of a symettric matrix and a skew symettric matrix.
That is what Laurent did. Just check that the way he wrote it is what you want.

Would I actually need to write the matrix out, though, or is it ok to leave it like that?

Originally Posted by Hellreaver

Would I actually need to write the matrix out, though, or is it ok to leave it like that?

Just write where (symmetric) and (skew symmetric).

Try starting with

So, we only have to prove that is

symmetric and that is skew symmteric.

Note that:



Now, you do it for the other case, , and you're pretty much done.

Well, the big reason I was asking is because I will need to be able to solve instances with actual values:

1 0
2 1

I need to represent this matrix as a sum of a symmetric matrix and a skew symmetric matrix.