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You might be interested to look at Section 21 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.

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.

You might be interested to look at Section 21 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 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.

You might be interested to look at Section 21 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.

You might be interested to look at Section 21 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: (simultaneously diagonalizable is in the sense of hermitian).

Consider 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.

You might be interested to look at Section 21 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 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.