.Let T:V->V be a linear map on a finite-dimensional real vector space.
Suppose A is the matrix representing T with respect to basis e.
Show that if A is diagonalizable then every matrix representation of T is diagonalizable.
With this question there is a hint that you may assume that any other matrix representation of T is of the form (X^-1)AX where X is an invertible matrix.
My thoughts for this question start with that if A is diagonalizable then there exists an invertible matrix X such that (X^-1)AX is diagonal. But I think I need to show that for all X (X^-1)AX is diagonalizable.
No: since is diagonalizable there exist an invertible matrix and a diagonal matrix s.t. .
Now, if is ANY other matrix representation of the map then there exists an invertible matrix s.t. , but then:
...and this means not only that is also diagonalizable but it will have the very diagonal form as
So I think that if (X^-1)AX is a matrix representation of T with respect to another basis then X must consist of the coefficients of the vectors in this basis as linear combinations of the vectors in e.
I think after this I have to show that (X^-1)AX has n linearly independent eigenvectors as this would imply diagonalizability but I'm not sure how to get there. Any help would be much appreciated