What is the practical application of the trace of a square matrix?

[emergentecon]

I'm interested in the use/application of the trace of a square matrix?
I am trying to get an intuitive feel for what it 'means' . . .

Along the lines of: for a 2x2 matrix, the determinant represents the area of the parallelogram.

I know it is the sum of the entries of the diagonal of a square matrix, but is there some practical, everyday, easy to understand, example of how the trace is used?

 

[emergentecon]

No one?

 

[homeomorphic]

I'm not sure about application. Some form of it comes up in quantum mechanics, but I'm not sure that's helpful here.

However, there are intuitive interpretations. You can think of the matrix as giving you a vector field, rather than a linear transformation. If you give the matrix a vector, which you can think of as a point in space, it spits out another vector if you multiply by it. Put that vector at the tip of the original vector. If you do that for all points in the plane, you get a vector field. The trace is then the divergence of this vector field, if you're familiar with divergence.

http://en.wikipedia.org/wiki/Divergence

Another way to interpret it is as some sort of derivative of the determinant, but that only works in special cases (in particular, for matrices that are obtaining by differentiating a one parameter family of matrices through the identity matrix).

If you have a diagonalizable matix, the trace is also the sum of the eigenvalues (counted with multiplicity). I'm thinking this should have some sort of physical significance, but it's hard to come up with a good example. Anyway, it's a good mathematical interpretation if your matrix happens to be diagonalizable.

Another thing about trace is that it's an invariant thing that doesn't depend on what coordinate system (or basis) you choose. So, one way you could use it, mathematically, is if I asked you if two matrices, A and B, were similar (ie. A = RBR^-1 or one is obtained from the other by changing the basis), if you calculated the trace and it came out different, you'd know they couldn't be similar.

 

[bhobba]

Its used extensively in QM, but expressed in the language of the bra-ket notation that is usual in that subject.

Fundamental to QM is the Born rule, which is if O is an observable (in QM an observable is a linear operator whose eigenvalues are the possible outcomes of an observation) then a positive operator of unit trace, P, exits, such that the expected value of the observation E(O) = trace(PO). By definition P is called the state of the quantum system.

I am sorry this is a bit advanced, but the trace is one of those things that doesn't have elementary applications - to the best of my knowledge anyway.

Thanks
Bill

 

[emergentecon]

Much appreciated guys!

 

[AlephZero]

If you have a diagonalizable matix, the trace is also the sum of the eigenvalues (counted with multiplicity). I'm thinking this should have some sort of physical significance, but it's hard to come up with a good example.

A different way to look at that is to consider the characteristic polynomial. For an $n \times n$ matrix, the trace is the coefficient of $x^{n-1}$.

This is important because the physics of a situation is always independent of the coordinate system used to describe it, therefore physical properties can be written in a coordinate-system-independent way as a function of the coefficients of the characteristic polynomial. For example in continuum mechanics of fluids the trace corresponds to the hydrostatic pressure, and the other coefficients are also useful to describe the onset of nonlinear behavior like plasticity. See http://en.wikipedia.org/wiki/Von_Mises_yield_criterion for example.

To model a problem numerically, you have to work in some particular coordinate system, but this sort of consideration means the physically important results are independent of the coordinate system you chose.