If A,B and B,C and C,A are simultaneously diagonalisable matrices then is this true that A,B,C are simultaneously diagonalisable? Here $A,B,C\geq{0}$.
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What does $A, B, C \geq 0$ mean? – Jim Feb 24 '15 at 08:16
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A,B,C are positive definite – Snehasish Bose Feb 24 '15 at 09:09
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Yes.
A set of diagonalizable matrices is simultaneously diagonalizable if and only if that set commutes (see, for example, this question). By definition a set of matrices commutes if each pair commutes, thus a set of diagonalizable matrices is simultaneously diagonalizable if and only if each pair in the set is simultaneously diagonalizable.
