# Matrix question?

Let C be a ivertible matrix. Define H(C) as a set of all matrices X that satisfy X^T C X = C(X^T is a rtansposed matrix).

Describe the set K(C) = {Y | H(Y)=H(C)}

Relevance

Suppose A and B are in K(C)

Xt (A+B) X = (Xt A + Xt B)X = XtAX + XtBX = (A+B)

Suppose A is in K(C)

Xt (a*A) X = a Xt A X = a*A

So K(C) is a space. (only C has to be invertible, there is nothing that sayis everything in K(C) has to be invertible).

Furthermore I agree whith what the previous poster said. Xt C X will be an element of K(C) for all X in H(C).

To see this just notice Xt (Xt C X) X = Xt (C) X . Furthermore Xt (Xt Xt C X X) X = Xt(Xt(C)X)X=XtXtCXX. So for any integer N (Xt)^N C X^N will be in K(C) by induction. But you can realise that X^N is also in H(C) by this kind of reduction so we can say:

Hence any linear combination of Xt^N C X^N for X in H(C) will be in K(C); but that does not actually give us anthing more then just keeping N=1 nince X^N is also another element of H(C).

Suppose that Y can not be expressed as such a linear combination. Then we can express it as L + (Y-L) where L is such a linear combination. Then for X in H(C); Xt (L + (Y-L))X = Xt L X + Xt(Y-L) X = L + Xt(Y-L)X = L + (Y-L)' where (Y-L)' !=(Y-L)

Suppose (Y-L)' can be expressed as a linear combination like described above. Lets call it L', now to reflect that representation. But then L' must be in K(C) so H(L')=H(C). This implies that Xt L' X = L' for X in H(C). But all X in H(C) must have full rank since they maintain the invertibility of C. So they are invertible (so is Xt) so:

L' = (Xt)i L' Xi = (Xi)t L' Xi . So L' is in K(C) (If X is in H(C) then so is Xi, by same steps a just here except with C instead of L'), But then L'=Xt L' X and as we saw L' = Xt (Y-L) X. This is not possible for invertible X.

So the only things in K(C) are linear combinations of Xt C X for X in H(C).

• if H(Y) = H(C),

then

X in H(Y) means that X^T Y X = Y

and

X in H(C) means that X^T C X = C

1. K(C) is not empty, since C is an element of K(C)

2. also, X^T C X is in K(C) if X is in H(C)