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### Understanding “matrices commute iff they share a common basis of eigenvectors”

I am trying to understand the answer to this question. It was pointed out that this doesn't necessarily hold for non-diagonalizable matrices, so the top answer starts by assuming $A$ and $B$ are ...
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### Do commuting matrices share the same eigenvectors?

In one of my exams I'm asked to prove the following Suppose $A,B\in \mathbb R^{n\times n}$, and $AB=BA$, then $A,B$ share the same eigenvectors. My attempt is let $\xi$ be an eigenvector ...
41 views

### Dimension of the space of matrices which is commutative to a given matrix.

Suppose I have a matrix $A$ in the space $V$ of $n$ by $n$ matrices. Then it is quite clear that $S=\{B : AB=BA\}$ form a subspace. I want to find out its dimension. I think it depends on the ...
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### Self-Adjoint Operators Basis of Eigenfunctions [duplicate]

I encountered the following statement in the context of the Hecke algebra on the space of cusp forms, left without proof: Let there be a family of commutative self-adjoint operators on a finite ...
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### If $A$ and $B$ are commuting Hermitian matrices, then they have the same eigenvectors?

If $AB = BA$ and both $A$ and $B$ are Hermitian matrices, then I can show that if $Av = \lambda v$, then $ABv = BAv = B\lambda v = \lambda Bv$. So $Bv$ is an eigenvector of $A$ as well. Where I am ...
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### To show orthogonal basis exist consisting common eigenvectors

Let $h$ be a positive definite hermitian form on $E$ and $A,B: E\rightarrow E$ be two hermitian endomorphisms which commute, $AB = BA$. Prove that there exists a orthogonal basis for $E$ consisting of ...
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### Show that every eigenvector of A is an eigenvector of B

Let $A, B\in\mathcal M_n(\mathbb{R})$ such that $AB=BA$ with $n$ distinct eigenvalues. 1) Show that if $v\in\mathbb{R}^n$ and $\lambda\in\mathbb{R}$, $Av=\lambda v\implies ABv=\lambda Bv$ 2) ...
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### Regarding a proof of: “if $A,B \in M_n(\mathbb{k})$ are diagonalizable and commute, they are simultaneously diagonalizable”.

As the title states, I'm looking for a proof of the following, Proposition. Let $A, B \in M_n(\mathbb{k})$ be commuting diagonalizable matrices, so that $AB = BA$. Therefore, $A$ and $B$ can be ...