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Suppose $A$ and $B$ are $n\times n$ matrices and they are both invertible. Can we say that $AB$ and $BA$ have the same eigenvalues?

ogulcan
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2 Answers2

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A more general result:

If $A$ and $B$ are $n \times n$ matrices, then we have for a scalar $\lambda \ne 0$:

$ \lambda$ is an eigenvalue of $AB$ iff $ \lambda$ is an eigenvalue of $BA$ .

Proof: if $ \lambda$ is an eigenvalue of $AB$ , then there is $x \ne 0$ such that

$(*)$ $ABx= \lambda x$.

Let $y:=Bx$. Then $y \ne 0$ (otherwise we would get from $(*)$ that $ \lambda =0$ or $x=0$).

Now we have

$$ BAy=BABx=B(ABx)=B( \lambda x)=\lambda Bx = \lambda y.$$

It follows that $\lambda$ is an eigenvalue of $BA$.

Fred
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    If $A, B$ are square and $AB$ is invertible then $A$ and $B$ are both invertible, so you can remove the $\lambda \neq 0$ assumption. $0$ is an eigenvalue of $AB$ if and only if $0$ is an eigenvalue of $BA$. (Equivalently, $AB$ is invertible if and only if $BA$ is invertible.) – Michael May 03 '19 at 13:28
  • The proof should consider the case for $AB$,$BA$ having all eigenvalues zero. – SOFe May 17 '19 at 05:55
  • More generally, the above result holds true even when $\mathbf{A}$ and $\mathbf{B}$ are rectangular, as long as $\mathbf{AB}$ and $\mathbf{BA}$ are square matrices (possibly of different orders!). – Gokulakrishnan CANDASSAMY Apr 27 '23 at 04:34
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Yes, this is correct. They have the same characteristic polynomial. Look at Wikipedia.

  • could you prove that with using diagonalization of matrices? I tried to write down these matrices as a product of a matrix with eigenvectors and a digonal matrix with eigenvalues but couldn't prove that. – ogulcan Nov 24 '16 at 22:11
  • Who says that the matrices are diagonalizable? –  Nov 24 '16 at 22:14
  • no one:) Thank you for your answer!:) – ogulcan Nov 24 '16 at 22:15
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    Since the asker says $A$ is invertible, we can just use the equation $$BA = A^{-1} (AB) A$$ of your reference which shows that $AB$ and $BA$ are similar (conjugated to each other) and hence have the same eigenvalues. – Jeppe Stig Nielsen Nov 24 '16 at 23:18