Linked Questions

3
votes
1answer
681 views

Can anyone explain why commuting matrices share common eigenvector? [duplicate]

If two matrices $A$ and $B$ commute with each other, why would they share some eigenvector? Does that mean that an eigenvector for $A$ is also an eigenvector for $B$ and vice-versa?
0
votes
0answers
115 views

If $A,B$ are commuting diagonalizable complex matrices , then $A,B$ have a common eigen-basis ? [duplicate]

If $A,B$ are diagonalizable complex matrices of size $n$ such that $AB=BA$ , then is there a common eigen-basis of $A$ and $B$ ?
1
vote
1answer
46 views

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 ...
31
votes
7answers
27k views

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 ...
35
votes
3answers
5k views

Simultaneous Jordanization?

IN a comment to Qiaochu's answer here it is mentioned that two commuting matrices can be simultaneously Jordanized (sorry that this sounds less appealing then "diagonalized" :P ), i.e. can be brought ...
17
votes
3answers
5k views

If matrices $A$ and $B$ commute, $A$ with distinct eigenvalues, then $B$ is a polynomial in $A$

If $A\in M_{n}$ has $n$ distinct eigenvalues and if $A$ commutes with a given matrix $B\in M_{n}$, how can I show that $B$ is a polynomial in $A$ of degree at most $n-1$? I think first I need to show ...
8
votes
2answers
5k views

If $AB=BA$, show that $B$ is diagonalizable.

We were asked to prove the following: Let $ A $ be an $n \times n$ matrix with $n$ distinct real eigenvalues. If $AB=BA$, show that $B$ is diagonalizable. It was suggested I show that an ...
3
votes
1answer
2k views

How do I prove that in every commuting family there is a common eigenvector?

The proof given by my textbook is highly non-satisfying. The author adopted some magic-like "reductio ad absurdum" and the proof (although is correct) didn't reveal the nature of this problem. I made ...
0
votes
1answer
1k views

If $A$ and $B$ commute, show that they have a common eigenvector? [duplicate]

That is, if $AB = BA$. I can see that it is true if all the eigenvalues of $A$ or $B$ are distinct. Let $(\lambda, \textbf{x})$ be an eigenpair of $A$ (or whichever one has distinct eigenvalues). So ...
2
votes
2answers
132 views

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 ...
0
votes
3answers
153 views

Matrices that Commute with of a Specific matrix

Let $a$ and $b$ be real numbers. Considere the $2\times 2$ matrix \begin{equation*}A=\left[\begin{array}{cc}a&b\\-b&a\end{array}\right]. \end{equation*} What is the centralizer of the matrix $...
1
vote
3answers
129 views

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) ...
0
votes
2answers
205 views

Quantum mechanics, conmutative operators.

If two operators $A$ and $B$ commute then any eigenvector of $A$ is an eigenvector of $B$? I know that if that happens there is a basis in which the eigenvectors of $A$ and $B$ are equal, but I don't ...
3
votes
0answers
188 views

Set of commuting Matrices $\Rightarrow $ Common Eigenvector [duplicate]

I am trying to prove that if we have an arbitrary set of commuting matrices in $M_n(\mathbb C)$ then they have a common eigenvector. Well, if we have only 2 matrices, the answer is easy and it has ...
3
votes
0answers
181 views

It is said that two matrices commute “if and only if” they share eigenvectors

It is for homework. But I am not asking the answer, since it seems to be already given in many posts like this one... I am pretty sure that I will get an A just copying and pasting it. My question is ...

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