Similar Matrices, Trace and Norm

Trace and Norm

The trace of a square matrix is the sum of its diagonal elements.

Let A and B be n×n matricies and consider the upper left entry of A*B. This is the sum of A1,j×Bj,1, as j runs from 1 to n. The ith entry in the diagonal is the sum of Ai,j×Bj,i. To find the trace of A*B, take the sum as i runs from 1 to n. This produces the sum of Ai,j×Bj,i, for all i and j between 1 and n. Yet this is symmetric with respect to A and B. We are merely rearranging the terms of the sum. Therefore the trace of A*B is the same as the trace of B*A.

If Q is a nonsingular matrix, consider the trace of QM/Q. This is the same as the trace of M/Q times Q, or M. The trace of a matrix is equal to the trace of any similar matrix. In other words, trace is basis invariant.

Given a matrix M, let T be the upper triangular matrix that is similar to M. You may need to expand the underlying field to do this, e.g. extend the reals up to the complex numbers. Once this is done, T can always be derived.

Recall that the determinant of T is the product of the diagonal elements. Subtract the variable s from these elements and multiply the monomials together to get the characteristic polynomial of T. The roots of this polynomial are, of course, the elements on the main diagonal. Thus the diagonal contains the eigen values, including multiplicities.

The trace of T is the sum of the diagonal elements, which happens to be the sum of the eigen values. Since eigen values are basis invariant, M has the same eigen values. We already showed M has the same trace as T. Therefore the trace of M equals the sum of its eigen values. You can add eigen values or diagonal elements; the result is the same.

The norm of a matrix is the product of its eigen values, including multiplicities. Again, the eigen values are basis invariant, so switch to a similar, upper triangular matrix T. The norm is now the product of the diagonal elements of T. This is also the determinant of T, which is the same as the determinant of M. So the norm is the determinant, or the product of the eigen values.

These definitions, trace and norm, are special cases of more general definitions that apply to endomorphisms on free/projective modules. If you study modules, you'll run into these more general definitions, but they are beyond the scope of linear algebra.