why is this true? I can't comprehend how he can just replace

>>8515054
Multiplying by a matrix then multiplying by it's inverse yields the original result. You're basically doing a transformation to a different coordinate system then transforming back.

>>8515061
but that would mean that all similar matrices are just the same matrix, right? That's obviously not true.

A is similar to B if there exists an S for which the following holds true

A = S^-1 B S

>>8515080
>but that would mean that all similar matrices are just the same matrix, right? That's obviously not true.
No, that would mean that all similar matrices have the same set of eigenvalues, which is true (the original image is the proof of that).

det(M^-1 xI M)=det(M^-1)det(xI)det(M)=(1/det(M))det(xI)det(M)= det(xI)

>>8515094
[eqn]det(M{^-1} xI M)=det(M^{-1})det(xI)det(M)=(\frac{1}{det(M)})det(xI)det(M)= det(xI)[/eqn]

I don't really know what xI is in your equation, but if x is a vector and I is the identity matrix, then it doesn't really make sense

>>8515099
Clearly x is a scalar.

>>8515092
I'm trying to understand the proof but I think I'm lacking some basic knowledge about this

if A = S^-1 B S
and S^-1 B S = B

then how come it doesn't follow that A = B?

>>8515099
but det(A-B) isn't equal to det(A) - det(B)

>>8515105
yeah that makes sense actually, also, the equation i just posted doesnt really answer the question since det is not additive.

But if x is a scalar, then obviously, by definition of identity and defining property of vector spaces [eqn]M^{-1}xIM = xM^{-1}IM=xIM^{-1}M=xI\cdot I=xI[/eqn]

>>8515112
yeah true, didnt think of that

>>8515109
ah, I see.

so in the case of xI, S^-1 xI S = xI holds true because xI is obviously diagonal.

Makes sense, thanks man.

>>8515054
det(MN)=det(M)det(N)
det(M)^-1=det(M^-1)
det(I)=1
So take any invertible matrix and multiply it on both sides of another matrix and the determinant doesnt change.

>>8515121
is this true now or is it not? Why'd you delete your post

>>8515117
this is great, thanks.

>>8515054
I commutes with every matrix, which makes it possible to write xI = xIM^{-1}M = M^{-1}xIM

what is this aiming to prove? something about changes of basis and eigenvalues ?

>>8515134
it's from here http://math.stackexchange.com/questions/87699/elegant-proofs-that-similar-matrices-have-the-same-characteristic-polynomial

>>8515138
cool thanks.
i always forget the most basic stuff about linear algebra