How do eigenvalues of a matrix X change if we linear transform the matrix X?

Question

I have a matrix $X$ which has eigenvalues $U$.
Now create a new matrix $Y = AX$ where $A$ is a nonsingular matrix.

How do the eigenvectors and eigenvalues of $Y$ change in relation to the eigenvectors and eigenvalues of $X$ and the matrix $A$?

Answer 1

If the eigenvector equation for $X$ is $$ Xv=\lambda v $$ then $$ Yv=AXv=A\lambda v=\lambda Av $$ so it comes down to what effect $A$ has on $v$ which will vary according to $A$.

In special cases, such as $A$ being a constant diagonal matrix with value $k$ on the diagonal then the above equation continues $$ \lambda Av=\lambda k Iv = \lambda kv $$ In such a special case the eigenvectors of $X$ and $A$ are the same and the eigenvalues are multiplied by $k$.