Question
Let $A$ and $B$ be $n \times n$ matrices. Prove that the matrix products $A B$ and $B A$ have the same eigenvalues. Hint: How should the eigenvectors be related?
Step 1
We need to show that the matrices $AB$ and $BA$ have the same eigenvalues. Recall that an eigenvalue $\lambda$ of a matrix $M$ is a scalar such that there exists a nonzero vector $v$ (called an eigenvector) satisfying $Mv = \lambda v$. Show more…
Show all steps
Your feedback will help us improve your experience
Victor Salazar and 59 other educators are ready to help you.
Ask a new question
Labs
Want to see this concept in action?
Explore this concept interactively to see how it behaves as you change inputs.
Key Concepts
Recommended Videos
Let $A$ and $B$ be $n \times n$ matrices, each with $n$ distinct eigenvalues. Prove that $A$ and $B$ have the same eigenvectors if and only if $A B=B A$
Show that if $A$ and $B$ are two $n \times n$ matrices, then the matrices $A B$ and $B A$ have the same characteristic polynomial, and thus the same eigenvalues (matrices $A B$ and $B A$ need not be similar though; see Exercise 55 ). Hint: \[ \left[\begin{array}{cc} A B & 0 \\ B & 0 \end{array}\right]\left[\begin{array}{cc} I_{n} & A \\ 0 & I_{n} \end{array}\right]=\left[\begin{array}{cc} I_{n} & A \\ 0 & I_{n} \end{array}\right]\left[\begin{array}{cc} 0 & 0 \\ B & B A \end{array}\right] \].
Eigenvalues and Eigenvectors
More on Dynamical Systems
Transcript
Watch the video solution with this free unlock.
EMAIL
PASSWORD