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This question already has an answer here:

I've often seen this property used when dealing with covariance matrices, which have just have every desirable property imaginable (e.g. they are symmetric, contain only real numbers, and are positive semi-definite), but I don't recall every seeing any use of this property outside of this context. Is it just a useful property of covariance matrices? Or is it a result of some of their non-unique properties, and if so, which properties?

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marked as duplicate by Matthew Towers, B. Mehta, Rahul, Community Apr 10 '18 at 21:58

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You didn't formulate your question correctly (there is no matrix as in the question, simple exercise: let $y$ be a multiple of $x$).

On the other hand, trying to understand what is your real question you might want to consider normal operators: note that there is always a basis formed by orthogonal eigenvectors (so you have an equivalence to the property that you seem to be considering).

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  • $\begingroup$ All $1\times 1$ matrices with coefficients in $\mathbb{F}_2$ have the property they originally stated. $\endgroup$ – user550675 Apr 10 '18 at 20:42
  • $\begingroup$ Never heard of covariance with other fields... :) $\endgroup$ – John B Apr 10 '18 at 20:43
  • $\begingroup$ That doesn't change that the claim that there is no such matrix is false as written. $\endgroup$ – user550675 Apr 10 '18 at 20:49
  • $\begingroup$ I suggest that you give your own answer. $\endgroup$ – John B Apr 10 '18 at 20:52

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