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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 ...
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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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### 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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### If $\bigoplus\limits_{i=1}^{k} Eig(A,\lambda_i)=\bigoplus\limits_{i=1}^{k} Eig(B,\mu_i)$ then $A$ and $B$ commute

Let $A,B$ be two $n\times n$-matrices over $\mathbb K$ and assume $\bigoplus\limits_{i=1}^{k} Eig(A,\lambda_i)=\bigoplus\limits_{i=1}^{k} Eig(B,\mu_i)=\mathbb K^n$, where $\lambda_i, \mu_i$ are ...
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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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