Suppose $A, B \in R^{n \times n}$ are symmetric, positive-definite. Can we say that they commute? If not, what additional conditions are required?
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1No you can't say they commute in general, you should be able to find $2 \times 2$ counterexamples very quickly. However, since you can always diagonalize such matrices they commute iff they are simultaneously diagonalizable. – Rammus Nov 27 '20 at 16:16
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@Rammus, does simultaneously diagonalizable mean that they have all eigenvectors in common? – user67724 Nov 27 '20 at 16:27
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yes, it is enough to have one of them positive. – J.E.M.S Mar 24 '23 at 15:00