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    • Thread Starter

    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 A=A^T, there exists a real diagonal \Lambda and an orthogonal S such that A= S \Lambda S^{T}
    • 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 B^T B \geq 0 for any B. I'm required to prove the converse using the ST - if A \geq 0 then A=B^T B.

    So far all I've really got is that since A is pos semidef, its quad form Q_A(\vec{u}) = \vec{u^T}A\vec{u}... 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?

    Have a look here: http://en.wikipedia.org/wiki/Cholesky_decomposition
    • Thread Starter

    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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Updated: March 25, 2011
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