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Spectral theorem quick questions watch

1. 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?
2. Have a look here: http://en.wikipedia.org/wiki/Cholesky_decomposition
3. Ah... it seems that a part of the spectral theorem has been taken out of the course... I'll read up on it anyway.

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