I've got a question on a paper here that's asking me to 'state' the spectral theorem. Bit of a broad question, really.
My current understanding of it is that it basically states:
- All eigenvalues of symmetric matrices are real
- all of its eigenvectors are also real, and are also pairwise orthogonal
Additional theorems that seem to stem from the Spectral Theorem (in my notes) are:
- For , there exists a real diagonal and an orthogonal S such that
- The inertia is given by the number of positive, negative and 0 evalues respectively.
- A symmetric matrix A is positive (semi)definite if and only if its eigenvalues are positive (non-negative).
Since my course obviously doesn't delve too far into the Spectral Theorem, what exactly am I 'stating' when I'm asked to state the ST? Are the first two points sufficient? I'd assume the latter 3 would require proofs and therefore aren't necessarily a part of the ST.
Also, separate question but on the same topic: I'm given that for any B. I'm required to prove the converse using the ST - if then .
So far all I've really got is that since A is pos semidef, its quad form ... but that's going to get me nowhere. Unless I could use the fact that its eigenvalues are positive...? Can each A be written in terms of S which is the matrix composed from its eigenvalues?
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Spectral theorem quick questions watch
- Thread Starter
Last edited by wanderlust.xx; 25-03-2011 at 19:45. Reason: Simpler title.
- 25-03-2011 19:43
- 25-03-2011 21:04
Have a look here: http://en.wikipedia.org/wiki/Cholesky_decomposition
- Thread Starter
- 25-03-2011 21:36
Ah... it seems that a part of the spectral theorem has been taken out of the course... I'll read up on it anyway.