By explicit construction of the matrices in question, show that any matrix T can be written as the sum of matrices:
a) symmetric matrix S and antisymmetric matrix A
b) real matrix R and imaginary matrix M
c) hermitian matrix H and skew-hermitian matrix K

Question

By explicit construction of the matrices in question, show that any matrix T can be written as the sum of matrices: 
a)  symmetric matrix S and antisymmetric matrix A
b) real matrix R and imaginary matrix M
c) hermitian matrix H and skew-hermitian matrix K

jamesj · Answer

are you stuck on this, really?  Which bit?  part b is trivial yes?

unklerhaukus · Answer

i have the answer i just dont find it particularly illuminating

jamesj · Answer

Hmm ... for part a it's

A = (1/2) (A + A^t) + (1/2) (A - A^t)

I think that's quite intuitive.

jamesj · Answer

In fact, all three parts have the 'same' answer if you replace
transpose
for
conjugate
for
complex transpose

jamesj · Answer

It's analogous also to writing a real function as the sum of an even and an odd function.

unklerhaukus · Answer

the answer for a) i have is 
$$S=\frac{1}{2}(T+\tilde{T}); \quad A=\frac{1}{2}(T-\tilde{T})$$

jamesj · Answer

right, where ~ means transpose

unklerhaukus · Answer

yeah

unklerhaukus · Answer

ok well is there a step before this that i can not see for some reason

jamesj · Answer

Um, I arrived at it by remembering that A + A^t is always symmetric.  Then I just had to scale the thing appropriately and find it's complement.  

Then I remembered the pattern works everywhere

jamesj · Answer

*its

unklerhaukus · Answer

So i guess my question should be why is $$A +\tilde {A} $$aka$$A+A^T$$ necessarily symmetric

jamesj · Answer

Because (A^t)^t = A, hence .... see it now?

unklerhaukus · Answer

$$A+A^T=(A^T+A)^T$$$$A^T+A = (A^T+A)^T$$

unklerhaukus · Answer

the bottom right corner i mean to have$$(A+A^T)^T$$

unklerhaukus · Answer

and the point is transposes do noting to the sum.

unklerhaukus · Answer

have i got it?

jamesj · Answer

Yes.  

A little more abstractly, the operation of taking the transpose is order 2.

This is why the same method will also work with the other operations.

jamesj · Answer

Remember when we were kids in primary school we'd make painting of butterflies

jamesj · Answer

We'd put a vertical fold in the paper and put some paint on both sides.  Then we'd fold the two sides together and open it up again.

The analogy isn't perfect, but that's sort of what's going on here.

jamesj · Answer

On that colourful note, I'm out of here.

unklerhaukus · Answer

thanks i think it makes more sense to me now

unklerhaukus · Answer

this is a bit like odd and even functions isnt it