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For two $n \times n$ nonnegative definite Hermitian matrices $A$ and $B$ over the real number field $\mathbb R$:

Question1:Is there always a nonsingular matrix $P$ over the same field $F$ which makes $P^{H}AP$ and $P^{H}BP$ both diagonal?

If the answer is yes, in which part of the matrix theory(or L.A.) can I find such results? If the answer is no, whether there's a similar result or some counterexample?

Question2:What special property does this kind of $P$ have?

Will it still be Hermitian or have some rank inequality in relation with the $rank(A)$ and $rank(B)$, like this sort.

Question3:What is its corresponding background in Abstract algebra?

For example, the Jordan decomposition correspond to the primary decomposition, like this sort.

Moreover I want to know if this result has any application?

Most probably in Lie algebra, because this proposition is taken from some Lie algebra course.

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  • $\begingroup$ I'm very curious, how such a question was raised and left unanswered during a Lie-algebra course... $\endgroup$
    – Henry.L
    Commented May 3, 2013 at 11:03
  • $\begingroup$ This question doesn't make sense as stated, since "nonnegative definite" and "Hermitian" aren't defined for a general field. (Even if your field is the complex numbers, do you want to assume $A$ and $B$ commute, so you can take $P$ to be unitary?) See mathoverflow.net/questions/118680 for information about which fields have the property that every symmetric matrix is diagonalizable. $\endgroup$
    – Henry Cohn
    Commented May 3, 2013 at 13:22
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    $\begingroup$ What do you mean by "non-negative definite" matrix over a general field? What does "Hermitian" mean over a general field? What "special property" do you have in mind. (See if you can identify such property in the case when $F={\mathbb R}$.) $\endgroup$
    – Misha
    Commented May 3, 2013 at 13:26
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    $\begingroup$ Have you read en.wikipedia.org/wiki/… ? $\endgroup$
    – S. Carnahan ♦
    Commented May 4, 2013 at 5:02
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    $\begingroup$ As far as I understand your question (which make little sense even after corrections: e.g. what is $F$ now?), it is basic material which you will find in any book treating these questions. Voting to close. $\endgroup$
    – Benoît Kloeckner
    Commented May 4, 2013 at 15:38

1 Answer 1

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You might be interested to look at Section 20 on Simultaneous diagonalization of a pair of Hermitian forms of the following book:

Prasolov, V. V. Problems and theorems in linear algebra. Translations of Mathematical Monographs, 134. American Mathematical Society, Providence, RI, 1994.

Among other things the following results are proved: (diagonalizable is in the sense of hermitian).

Consider $n\times n$ hermitian matrices $A$ and $B$.

(1) If $A$ is a positive definite then $A$ and $B$ are simultaneously diagonalizable.

(2) If $A$ is invertible then $A$ and $B$ are simultaneously diagonalizable if and only if $A^{-1}B$ is diagonalizable and all its eigenvalues are real.

(3) If $A$ and $B$ are both nonpositive or nonnegative then $A$ and $B$ are simultaneously diagonalizable. (It seems that, this is the result you are looking for).

(4) If $A$ and $B$ are not simultaneously isotropic then $A$ and $B$ are simultaneously diagonalizable.

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