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Quick questions:

  • if you have a diagonal matrix $A$ and a unitary matrix $B$. Do $A$ and $B$ commute?

  • if $A$ and $B$ are positive definite matrices. if $a$ is an eigenvalue of $A$ and $b$ is an eigenvalue of $B$, does it follow that $a+b$ is an eigenvalue of $A+B$?

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No and no (that is, not always). Have you tried any examples? –  Qiaochu Yuan Apr 30 '12 at 20:25
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2 Answers

up vote 4 down vote accepted

For the first question, the answer is no, an explicit example is given by $A:=\pmatrix{1&0\\ 0&2}$ and $B=\pmatrix{1&1\\ -1&1}$. An other way to see it's not true is the following: take $S$ a symmetric matrix, then you can find $D$ diagonal and $U$ orthogonal (hence unitary) such that $S=U^tDU$ and if $U$ and $D$ commute then $S$ is diagonal.

For the second question, the answer is "not necessarly", because the set $$\{a+b\mid a\mbox{ eigenvalue of }A,b\mbox{ eigenvalue of }B\}$$ may contain more elements than the dimension of the space we are working with.

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(Just as a little resource, \mid is probably the best markup for the vertical bar in set notation.) –  rschwieb Apr 30 '12 at 22:49
    
@rschwieb I agree, especially when we don't have division. –  Davide Giraudo May 1 '12 at 8:25
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Perhaps you know of so-called normal matrices, those complex matrices $M$ that commute with their transpose conjugate $M^{\dagger}$? If the answer to your first question were yes, there would be only "trivial", i.e. diagonal, normal matrices. Indeed, any normal matrix is unitarily equivalent to a diagonal one (all normal endomorphisms of a hermitian vector space are diagonalisable in an orthonormal basis).

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