• Theorem:
  Let A and B be two complex-valued matrices of dimensions n×m and m×n, respectively.
  Then both AB and BA are defined and
  tr(AB) = tr(BA)

Proof: The element in the j-th row, j-th column of AB is given by

Σ_ka_j,kb_k,j

The trace of AB is then

Σ_jΣ_ka_j,kb_k,j
where k runs from 1 to m and j runs from 1 to n.

On the other hand, the element in the k-th row, k-th column of BA is

Σ_jb_k,jb_k,ja_j,k

and the trace of BA is

Σ_kΣ_jb_k,jb_k,ja_j,k

But the order of summation can be reversed and also the order of the terms in the product can be reversed. When this is done the result is:

Σ_jΣ_ka_j,kb_k,j
where k runs from 1 to m and j runs from 1 to n.

This is exactly the same as the expression for the trace of AB.

The Trace of a Matrix and Its Eigenvalues

First a simple version of the proposition will be considered. It is not necessary to consider this case separately but it makes the proof of the theorem easier to absorb. Let M be a complex-valued n×n matrix that is diagonalizable; i.e., there exists V such that

V^-1MV = Λ

where Λ is the diagonal matrix of the eigenvalues of M.

Now consider V^-1MV as (V^-1M)V. Then the previous theorem applies so

tr(Λ) = tr(V^-1MV) = tr((V^-1M)V) = tr((V·V^-1)M) = tr(I·M) = tr(M)

Thus the sum of the eigenvalues of a diagonalizable matrix is equal to its trace. A matrix M is diagonalizable if all of its eigenvalues are different; i.e., the multiplicity of every eigenvalue is 1.

When the multiplicities of some of a matrix's eigenvalues of greater than 1 it is not diagonalizable but instead for any matrix A there exists an invertible matrix V such that

V^-1AV = J

where J is of the canonical Jordan form, which has the eigenvalues of the matrix on the principal diagonal and elements of 1 or 0 mext to the principal diagonal on the right and zeroes everywhere else.

As in the simple case of diagonalizable matrices considered before

tr(J) = tr(V^-1AV) = tr(VV^-1A) = tr(I·A) = tr(A)

The canonical Jordan form matrix J has the eigenvalues of A on its principal diagonal so the trace of J is equal to the sum of the eigenvalues of A. Thus

tr(A) = Σλ_j