Is there a way to calculate determinant of eigenmatrix of a matrix w/o calculating eigenmatrix? By eigenmatrix, I mean a diagonal matrix with elements of diagonal being coordinates of eigenvector of said matrix.
While discussing this with someone, I was told "determinants are invariant under similarity". What does that mean?
The determinant of the matrix is the product of the eigenvalues of that matrix.
If $A $ is diagonisable by some diagonal $D $, ie $A=PDP^{-1} $, then by properties of the determinant $det (A)=det (D)$, taking the determinant of both sides of the equation.
This obviously works if instead of $D $ we have another matrix that is similar to $A $: $A=PBP^{-1} $ implies $det (A)=det (B)$, and the determinant of $A $ is invarient, ie $det (A)=det (D)=det (B)$.