Thread: Matrices represented by Symmetric/Skew Symmetric Matrices

1. 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?

2. 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: $A=\frac{1}{2}(A+A^T)+\frac{1}{2}(A-A^T)$, where $A^T$ is the transpose matrix of $A$.

3. 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...

4. 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.

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

6. 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 $A = B + C$ where $B = \tfrac{1}{2}(A+A^T)$ (symmetric) and $C = \tfrac{1}{2}(A-A^T)$ (skew symmetric).

7. Try starting with $A=\frac{1}{2}(A+A^{T})+\frac{1}{2}(A-A^{T})$

So, we only have to prove that $\frac{1}{2}(A+A^{T})$ is

symmetric and that $\frac{1}{2}(A-A^{T})$ is skew symmteric.

Note that:

$\frac{1}{2}(A+A^{T})^{T}=\frac{1}{2}(A^{T}+(A^{T}) ^{T})=\frac{1}{2}(A+A^{T})$

Now, you do it for the other case, $\frac{1}{2}(A-A^{T})$, and you're pretty much done.

8. 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.