Thread: Matrix Eigenvalues

Can anybody please tell me what is the real physical meaning of an eigenvalue of any square matrix?

That depends heavily on the real physical meaning of the matrix.

give me an example

well, i want to know what was he motivation behind the "invention" of eigenvalue concept

I can't really give any concrete examples, but I can at least point to this.

I can, I think, though I am way off my patch here.

It seems to be an axiom of quantum physics that any observable event in nature can be represented by a linear operator on some vector space, and that the result of any physical measurement of the action of on this space, specifically an Hilbert space , will by an eigenvalue for the operator , Or rather, since we are the quantum world, the spectrum (not necessarily non-degenerate) of eigenvalues for give the range of discrete values for these measurements,

Back on my own patch: it is a nice (and not difficult) proof that the eigenvalues of an Hermitian operator (the physicists choice of operator) are always real (try it!), which makes complete sense with the above: what is meant by an imaginary measurement???

Thanks guys. I didn't quite understand the quantum mech. definition, but the wikipedia link helped.

Originally Posted by PritishKamat

Thanks guys. I didn't quite understand the quantum mech. definition, but the wikipedia link helped.

Be careful here. In quantum mechanics you are not working with a matrix but with a linear operator, often unbounded. The notion of a spectrum in that case becomes quite a bit more complicated.

In finite dimensions all elements of the spectrum are eigenvalues. And all matrices are representations of bounded (continuous) linear operators, once a basis has been fixed. In that case an eigenvector is one on which the linear transformation works as a simple stretching, contraction or reversal and the eigenvalue is the scale factor.