Is any real symmetric positive definite matrix similar to a triadiagonal symmetric positive definite matrix?

From Householder lemma it is known that if $B$ is symmetric, there exists a orthogonal matrix $H$ such that $C=H^\top B H$ is triadiagonal symmetric. The question is whether $C$ is definite positive; it has the same (positive) eigenvalues as $B$ so I would say yes.

Is this correct?

  • $\begingroup$ I think you are correct - as far as I know if a symmetric matrix has all positive eigenvalues then it is positive definite. But I don't know how to verify it. $\endgroup$ – Matt Dickau Oct 15 '15 at 16:06

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