## If Matrices Commute $AB=BA$, then They Share a Common Eigenvector

## Problem 608

Let $A$ and $B$ be $n\times n$ matrices and assume that they commute: $AB=BA$.

Then prove that the matrices $A$ and $B$ share at least one common eigenvector.

Let $A$ and $B$ be $n\times n$ matrices and assume that they commute: $AB=BA$.

Then prove that the matrices $A$ and $B$ share at least one common eigenvector.

Let $\lambda$ be an eigenvalue of $n\times n$ matrices $A$ and $B$ corresponding to the same eigenvector $\mathbf{x}$.

**(a)** Show that $2\lambda$ is an eigenvalue of $A+B$ corresponding to $\mathbf{x}$.

**(b)** Show that $\lambda^2$ is an eigenvalue of $AB$ corresponding to $\mathbf{x}$.

(*The Ohio State University, Linear Algebra Final Exam Problem*)

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Let $A$ and $B$ be an $n \times n$ matrices.

Suppose that all the eigenvalues of $A$ are distinct and the matrices $A$ and $B$ commute, that is $AB=BA$.

Then prove that each eigenvector of $A$ is an eigenvector of $B$.

(It could be that each eigenvector is an eigenvector for distinct eigenvalues.)

Add to solve laterLet $A$ and $B$ be $n\times n$ matrices.

Suppose that these matrices have a common eigenvector $\mathbf{x}$.

Show that $\det(AB-BA)=0$.

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