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I am trying to prove that a real symmetric matrix is diagonalizable. I have seen a few proofs in which the matrix is only symmetric, but I am hoping to avoid some of the heavy machinery that is used in those proofs (such as the Gram-Schmidt process). Any reference to a proof would also be much appreciated.

lalaman
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    Try reading about the all-important "Spectral Theorem", at least the version applied to real matrices. – DonAntonio May 09 '17 at 22:32
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    The Gram-Schmidt process is not heavy machinery! – Qiaochu Yuan May 09 '17 at 22:48
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    The process being used is sort of inevitably going to be along the lines of Gram-Schmidt, because the whole point of the proof is that if $A$ is symmetric and $v$ is an eigenvector then the orthogonal complement of the span of $v$ is invariant under $A$. (Also, you need the realness; in the complex case you have to deal with Hermitian matrices, which generally are not symmetric.) – Ian May 09 '17 at 23:48

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