The trace of square matrix?

Question

if A is nxn matrix of some basis |b>. How we can show that the trace of A is actually the sum of the diagonal elements?

I want to understand why Aii = < i |A| i >.

I know unitary matrix can transform basis to another like U|b> = |b'>. can this help to understand the concept of the trace?

I'm not a physics person so that might be straightforward to understand to others.

Answer 1

First, I think this question is better suited for the Physics Stack Exchange or the Math Stack Exchange because you're not asking a coding related question.

Second, the trace is literally defined as the sum of the diagonal elements of a matrix.

Third, your other question is about the inner product and Dirac notation. Dirac called the <a| "bra" and the |b> "ket". The inner product is basically the dot product but for higher dimensional space. In physics, Dirac notation is a shorthand we use to make writing matrix operations for quantum systems cleaner. You'll see bra's and ket's used to represent wave functions and spin states, and the inner product is then the energy of the system in that state and the spin of that state. Again, this question is really better suited for the Physics Stack Exchange.

Answer 2

A key property of trace is tr(BA) = tr(AB), which is readily proved from the definition as the sum of the diagonal elements. If you change basis, a matrix A will change into A~ = XAinv(X). So from the above tr(A~) = tr(A). Which is anpther way of saying that the trace is the sum of the diagonal elements, whaterver basisyou choose.

Answer 3

In a given basis, the ket |i> can be written as a column vector, with all elements 0 except the i^th, which is 1; likewise, the bra can be written as a row vector, with all elements 0 except the i^th, which is 1.

<i|A|i> = [Sum j][Sum k]<i|_j A_jk |i>_k
= [Sum j][Sum k] ∂_ij A_jk ∂_ki
= A_ii