Prove that trace is a similarity invariant that is if matrix

Prove that trace is a similarity invariant, that is if matrix B is similar to A show that tr (A) = tr (B).

Solution

Trace of a matrix is the sum of the diagonal elements of a matrix.

If A and B are similar matrix then an invertible matrix exists

P = A^-1BP

As per property : Trace (AB)= Trace(BA) for any nxn matrices A and B ( cyclic property)

Using this we have Tr(A) =Tr(A^-1BP) = Tr(BPP^-1) = Tr(B)

Hence Tr(A) = Tr(B)

=