It is the diagonal entries of the matrix raised to a power.

The trace of a 3 by 3 matrix A is defined as the summation of n=3;i=1;aii.

It is a Hermitian positive-semidefinite matrix of trace one that describes the statistical state of a quantum system.

Hermitian matrix is defined as A=A^(dagger). Meaning that NxN matrix A is equal to it's transposed complex conjugate.

Trace is defined as adding all the terms on the diagonal.

The trace of an nxn matrix is usually thought of as the sum of the diagonal entries in the matrix. However, it is also the sum of the eigenvalues. This may help to understand why the proof works. So to answer your question, let's say A and B are matrices and A is similar to B. You want to prove that Trace A=Trace B If A is similar to B, there exists an invertible matrix P such that A=(P^-1 B P) Now we use the fact that Trace (AB)= Trace(BA) for any nxn matrices A and B.This is easy to prove directly from the definition of trace. (ask me if you need to know) So using this we have the following: Trace(A)=Trace(P^-1 B P)=Trace (BPP^-1)=Trace(B) and we are done! Dr. Chuck

kncsJVKS;