Talk:Rayleigh quotient

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Interpretation from the eigenvalue equation[edit]

I found it very helpful to see that the Rayleigh quotient can be derived from the eigenvalue equation (left multiply by the (conjugate) transpose of x, then divide both sides by x*x). And thus the Rayleigh quotient is the approximate eigenvalue of an approximate eigenvector. Although this interpretation is quite trivial, I think it would fit Wikipedia well. — Preceding unsigned comment added by 94.210.213.220 (talk) 23:27, 7 December 2016 (UTC)[reply]

Special case of covariance matrices[edit]

Why can \Sigma be written as A'A?
Is A Hermitian?
Here are my thoughts: because \Sigma is a covariance matrix, it is positive semi-definite, and hence can be decomposed by Cholesky decomposition into A' and A, which are lower and upper triangular respectively. So the Cholesky decomposition gives A' and A which are not Hermitian, so why use the same letter A as above?
Does this apply to only covariance matrices and not all positive semi-definite symmetric matrices, or are they the same thing?
The following is not a sentence and needs help from someone who knows what is trying to be expressed: "If a vector x maximizes \rho, then any vector kx (for k \neq 0) also maximizes it, one can reduce to the Lagrange problem of maximizing [summation] under the constraint [summation]." Also, why the constraint?
There's a proof in here and I'm not sure why. It would be helpful if one would write explicitly what is being proved beforehand, and why the proof is being written.
This section needs an intro that gives it context with respect to other mathematical techniques, and it needs to explain why and what is being done.141.214.17.5 (talk) 16:20, 16 December 2008 (UTC)[reply]

I have tweaked the prose a bit, which I hope clarifies most of the points above.

In answer to one specific point: yes, one could apply the argument to any symmetric matrix M using its Cholesky factors L and L^T.

The specialisation here to covariance matrices is because this is a particularly important application: the argument establishes the properties of Principal Components Analysis (PCA), and its usefulness.

The empirical covariance matrix is defined to be A^TA (or to be exact, a linear scaling of it), where A is the data matrix, so that gives a natural starting point. Jheald (talk) 09:56, 17 June 2013 (UTC)[reply]

Shouldt this article be merged with Min-max theorem, which treats the same topic in more depth? Kjetil B Halvorsen 17:12, 11 February 2014 (UTC) — Preceding unsigned comment added by Kjetil1001 (talk • contribs)