What is the Connection Between Matrix Trace and Endomorphism?

[tamintl]

Consider the 4x4 matrices
A =
(1 2 3 4)
(5 6 7 8)
(9 10 11 12)
(13 14 15 16)B=
(1 2 3 4)
(8 5 6 7)
(11 12 9 10)
(14 15 16 13)

The question I was asked was the following: Show that there does not exist an endomorphism f: ℝ⁴ -> ℝ⁴ and basis 'a' and 'b' of R^4, such that A = _a[f]_a and B=_b[f]_b.

I have read in my notes and found that if the traces of the two matrices are not the same then they cannot represent the same endomorphism.

I am struggling to see the intuition behind this though.

Can anyone shed some light?

Many thanks

 

[Office_Shredder]

The trace of a linear transformation is the sum of the eigenvalues of the matrix, and so is independent of the choice of basis.

Alternatively, you can use the fact that Tr(AB) = Tr(BA) to show that if you conjugate a matrix by another matrix the trace is unchanged.