Is the determinant of a matrix independent of the basis in which that matrix is expressed? For example, when expressing in matrix form a quantum mechanics operator, or when comparing the same matrix but in its diagonal form.
Every case I try, I find it is independent of the basis. But why? How can I prove it, or what counter example could I find?
Thanks in advance.
On
A linear transformation from a (finite dimensional) vector space to itself has a determinant. Once you put a coordinate system on your space this linear transformation also has a matrix representation. You can use the entries of this matrix and a certain formula to calculate the determinant of the linear transformation (we call this "the determinant of the matrix", but most of the time that's a red herring).
The determinant was there before the coordinate system, and thus the determinant cannot in any conceivable way depend on the coordinate system.
Yes, it's a consequence of Binet's theorem: $\det(AB)=\det(A)\det (B)$. Therefore \begin{align}\det(PAP^{-1})&=\det (P)\det(A)\det(P^{-1})=\det(P)\det(P^{-1})\det(A)=\\&=\det(I_n)\det(A)=\det (A)\end{align}