Trace of a Matrix: Definition & Analysis

[lathawarrier]

In many books and also in wikipedia, the Trace of a matrix is defined as sum of its diagonal elements. For a general matrix, it does not make much sense, as any element is as important any other element. An alternative definition (in wikipedia for example) is that the Trace of the matrix is the sum of its eigenvalues. Such a definition makes lot of sense. For specialized matrices, for example diagonal matrices, trace (sum of eigenvalues) is the same as the sum of its diagonal elements. For a general matrix, the sum of the diagonal elements is not the sum of its eigen values and so is not the trace of the matrix.

Anyone disagrees?

 

[Bill_K]

The trace of a matrix is invariant under rotations (orthogonal transformations), so it has the same value whether you have already diagonalized the matrix or not.

 

[arash1r]

what physically means the Trace of a Matrix? dose it has a physical concept?

 

[bhobba]

In QM its value is from the standard trace formula where the expected value of an observable R is E(R) = Trace(pR) where p is the system state. System states are positive operators of unit trace.

The reason for the formula is very deep and is the subject of Gleasons theorem. It also follows from the assumption that expectation values are additive as proven by Von Neumann in his famous no go theorem - but that is a stronger assumption than made in Gleason's theorem. However both are evaded by hidden variable theories - but because of Von Neumann's reputation people initially didn't see it could be evaded - a few did but they were initially ignored and it had to wait for Bell and a few others to point it out. It was realized from the outset Gleason could be evaded - but the evasion is a bit trickier because Gleason implies Kochen-Specker.

The following goes into these issues in much more detail as well as gives proofs:
http://arxiv.org/pdf/quant-ph/0507182v2.pdf

Thanks
Bill

 

[tom.stoer]

For a general matrix, the sum of the diagonal elements is not the sum of its eigen values and so is not the trace of the matrix.

Anyone disagrees?

for quadratic N*N matrices I disagree b/c you can always diagonalize quadratic matrices, and the trace is an invariant w.r.t. diagonalization

 

[TGlad]

what physically means the Trace of a Matrix? dose it has a physical concept?

Well it is the sum of the ellipsoid lengths generated by applying the matrix to a sphere (of the appropriate number of dimensions).

Just as the determinant gives the volume of the ellipsoid (the product of the lengths), the trace gives the sum of the lengths.

You could think of it as a different form of average for the 'size' of the matrix.

 

[TGlad]

Actually I think my last explanation only applies to symmetric matrices. Wikipedia suggests that rotation matrices have a trace 1 + 2*cos(angle)... is this right?

 

[Einstein Mcfly]

What about the product of the eigenvalues of the density matrix? As you turn up the temperature of a thermal ensemble and the population spreads, the trace is preserved by the product of the eigenvalues changes. Is there any way to use this to tell you something useful about the ensemble apart from the obvious limits?

 

[tom.stoer]

For a density matrix you have

\rho = e^{-\beta H}
\rho_{mn} = \langle m|\rho|n\rangle = \langle m|e^{-\beta E_n} |n\rangle = e^{-\beta E_n}\,\delta_{mn}

\text{tr}\rho = \sum_n e^{-\beta E_n}

For the product you get

\prod_n e^{-\beta E_n} = e^{-\beta \sum_n E_n} \to 0

for typical systems b/c the sum diverges

 

[arash1r]

I have a question: when the poynting vector and energy propagation in electromagnetic waves are in the same direction and when are in different direction?