# Do matrices $AB$ and $BA$ have the same minimal and characteristic polynomials?

Let $A, B$ be two square matrices of order $n$. Do $AB$ and $BA$ have same minimal and characteristic polynomials?

I have a proof only if $A$ or $B$ is invertible. Is it true for all cases?

• The coefficients of the characteristic polynomial are continuous functions in the entries of a matrix, so if the characteristic polynomials of $AB$ and $BA$ coincide for a dense set of $A$ (or a dense set of $B$) then they always coincide. The coefficients of the minimal polynomial, on the other hand... Feb 22 '13 at 21:12
• @cmi obviously not. Try to figure out 2 different polynomials with the same set of roots. Is not hard. Jul 4 '19 at 7:41
• people.math.sc.edu/howard/Classes/700/charAB.pdf
– Bach
Aug 16 '19 at 19:32

Before proving $AB$ and $BA$ have the same characteristic polynomials show that if $A_{m\times n}$ and $B_{n\times m}$ then characteristic polynomials of $AB$ and $BA$ satisfy following statement: $$x^n|xI_m-AB|=x^m|xI_n-BA|$$ therefore easily conclude if $m=n$ then $AB$ and $BA$ have the same characteristic polynomials.

Define $$C = \begin{bmatrix} xI_m & A \\B & I_n \end{bmatrix},\ D = \begin{bmatrix} I_m & 0 \\-B & xI_n \end{bmatrix}.$$ We have \begin{align*} \det CD &= x^n|xI_m-AB|,\\ \det DC &= x^m|xI_n-BA|. \end{align*} and we know $\det CD=\det DC$ if $m=n$ then $AB$ and $BA$ have the same characteristic polynomials.

• I understood the proof, it's nice.. but is there any intuition abt why we consider C and D in such way? May 8 at 7:00

If $A$ is invertible then $A^{-1}(AB)A= BA$, so $AB$ and $BA$ are similar, which implies (but is stronger than) $AB$ and $BA$ have the same minimal polynomial and the same characteristic polynomial. The same goes if $B$ is invertible.

In general, from the above observation, it is not too difficult to show that $AB$, and $BA$ have the same characteristic polynomial, the type of proof could depends on the field considered for the coefficient of your matrices though. If the matrices are in $\mathcal{M}_n(\mathbb C)$, you use the fact that $\operatorname{GL}_n(\mathbb C)$ is dense in $\mathcal{M}_n(\mathbb C)$ and the continuity of the function which maps a matrix to its characteristic polynomial. There are at least 5 other ways to proceed (especially for other field than $\mathbb C$).

In general $AB$ and $BA$ do not have the same minimal polynomial. I'll let you search a bit for a counter example.

• 5 other ways? I'm quite curious what those ways are. I only know of the continuity argument and an argument involving determinant identities on block matrices. Would it be possible to provide a reference to some other methods?
– EuYu
Feb 22 '13 at 17:37
• @EuYu, I have no reference, sorry. Feb 22 '13 at 22:11
• @EuYu one other method is the argument that matrices $A=(a_{ij})$ and $B=(b_{ij})$ are invertible over the field $K(a_{ij}, b_{ij})$ Feb 15 '20 at 17:58
• See this short note by JH Williamson from 1953 cambridge.org/core/services/aop-cambridge-core/content/view/… Feb 15 '20 at 18:09

Hint: Consider $A = \begin{bmatrix} 0 & 1 \\ 0 & 0 \end{bmatrix}$ and $B = \begin{bmatrix} 0 & 0 \\ 0 & 1 \end{bmatrix}$. What do you get in that case?

• This shows that $AB$ and $BA$ have different minimal polynomial. But characteristic polynomials are the same right?
– user464147
Nov 25 '17 at 1:39

It's not true that their characteristic polynomials will be the same in the general case. The best result in this general vein is the following.

Let $A\in\mathbb{F}^{m \times n}$ and let $B\in\mathbb{F}^{n \times m}$, and $AB$, $BA$ with minimal polynomials (over $\mathbb{F}$) $m_{AB}(x)$ and $m_{BA}(x)$ respectively. Then one of the following holds:

$m_{AB}(x) = m_{BA}(x)$, or $m_{AB}(x) = x \cdot m_{BA}(x)$, or $x\cdot m_{AB}(x) = m_{BA}(x)$.

It's easy, just use the fact that $(BA)^k=B(AB)^{k-1}A$.

• in general means for m not equal to n?? Sep 16 '14 at 21:16
• The question says (and always has said) square matrices $A,B$. If you are answering a more general question, then you should announce this. Also, answering a more general question is only useful if this is no more difficult than the actual question, or if the more general solution sheds more light on the solution. Dec 23 '15 at 21:10

For square matrices, the characteristic polynomials are same, but for $$A$$ a matrix of size $$m \times n$$ and $$B$$ a matrix of size $$n \times m$$ we have $$x^{m}C_{BA}(x)=x^{n}C_{AB}(x)$$. This implies that the nonzero eigenvalue of $$AB$$, counted with multiplicities, are same as nonzero eigenvalue of $$BA$$.

That is, if $$A$$ is of size 7×4 and $$B$$ is of size 4×7 and assume that the 4×4 matrix $$BA$$ has nonzero eigenvalues 1,1,3 so fourth eigenvalue of $$BA$$ is 0. Then the 7×7 matrix $$AB$$ will also have nonzero eigenvalue 1,1,3 and remaining four eigenvalue of $$AB$$ are zero.

There are a lot of proofs for characteristic polynomials to be same. I want to provide mine. It may be more complicated, but it is less "consider magic product of matrices".

Let $$\chi_M(x)$$ denotes a characteristic polynomial $$\chi_M(x) = det(x - M)$$

Lets prove the fact: For square matrices $$A$$ and $$B$$ holds $$det(AB - x) = det(BA - x) \Leftrightarrow \chi_{AB}(x) = \chi_{BA}(x)$$.

If $$det(A) \neq 0$$ then it follows from $$det(AB - x) = det(A^{-1}A)det(AB - x) = det(A^{-1})det(AB - x)det(A) = det(BA - x)$$.

If $$det(A) = 0$$ there are finite number of such $$s \in \mathbb R$$ that $$\chi_A(s)=0$$ because $$\chi_A(s)$$ is a finite-degree polynomial. Then there are infinite number of such $$s$$ that $$\chi_A(s) \neq 0$$. For all such $$s$$ we know $$\chi_{(A-s)B}(x) = \chi_{B(A-s)}(x)$$ as a result of a previous case. For every fixed $$x$$ we see two finite-degree polynomials ($$x$$ is fixed, $$s$$ is variable) $$\chi_{(A-s)B}(x)$$ and $$\chi_{B(A-s)}(x)$$ which are equal in infinite number of points. Then we conclude they are equal at every $$s$$. At $$s = 0$$ we get the result $$\chi_{AB}(x) = \chi_{BA}(x)$$ at every $$x$$.

For square matrices we are done!

Key fact (proof below): If $$A$$ is $$m\times n$$, $$B$$ is $$n\times m$$ and $$n \geq m$$ then $$\chi_{BA}(x) = \lambda^{n-m}\chi_{AB}(x)$$.

Consider $$n\times n$$ matrices $$A' = \left(\dfrac{A}{0}\right)$$ and $$B' = (B\mid0)$$. We just put zero rows and columns to make matrices $$n\times n$$.

First, $$B'A' = BA \Rightarrow x - B'A' = x - BA \Rightarrow \chi_{B'A'}(x) = \chi_{BA}(x)$$

Second, $$A'$$ and $$B'$$ are square matrices. Then due to the fact above we have $$\chi_{B'A'}(x) = \chi_{A'B'}(x)$$.

Third, $$\chi_{A'B'}(x) = det(x - A'B') = det\begin{pmatrix}x - AB & 0 \\ 0 & \begin{matrix}x & 0 & \ldots & 0 \\ 0 & x & \ldots & 0 \\ \vdots & \vdots & \ddots & \vdots \\ 0 & 0 & \ldots & x \\\end{matrix}\end{pmatrix} = det(x - AB)x^{n - m} = x^{n-m}\chi_{AB}(x)$$

So, we see $$\chi_{BA}(x) = \chi_{B'A'}(x) = \chi_{A'B'}(x) = x^{n-m}\chi_{AB}(x)$$

Yes, $$AB$$ and $$BA$$ have the same characteristic polynomial.

Basic facts: $$\det(A^T) = \det(A)$$, $$\det(AB) = \det(A) \det(B)$$

1. $$A$$ and $$A^T$$ share the same characteristic polynomial.

\begin{align*} \det(xI-A) = \det((xI-A)^T) = \det(xI-A^T) \end{align*}

1. Similar matrices have the same characteristic polynomial. If $$B = PAP^{-1}$$,

\begin{align*} \det(xI - B) &= \det(xI - PAP^{-1}) \\ &= \det(P(xI - A)P^{-1}) \\ &= \det(P)\det(xI - A)\det(P^{-1}) \\ &= \det(xI - A) \end{align*}

1. Determinant of a block triangular matrix:

\begin{align*} \det \begin{pmatrix}A & B \\0 & C\end{pmatrix} = \det(A) \det(C) \end{align*}

Using block multiplication, please verify that $$\begin{pmatrix}I & -A \\0 & I\end{pmatrix} \begin{pmatrix}AB & 0 \\B & 0\end{pmatrix} = \begin{pmatrix}0 & 0 \\B & BA\end{pmatrix} \begin{pmatrix}I & -A \\0 & I\end{pmatrix}$$.

Therefore, the matrices $$\begin{pmatrix}AB & 0 \\B & 0\end{pmatrix}$$ and $$\begin{pmatrix}0 & 0 \\B & BA\end{pmatrix}$$ are similar, and have the same characteristic polynomial.

\begin{align*} \det\left[x\begin{pmatrix}I & 0 \\0 & I\end{pmatrix} - \begin{pmatrix}AB & 0 \\B & 0\end{pmatrix}\right] &= \det(xI - AB) \det(xI) \end{align*} \begin{align*} \det\left[x\begin{pmatrix}I & 0 \\0 & I\end{pmatrix} - \begin{pmatrix}0 & 0 \\B & BA\end{pmatrix}\right] &= \det(xI) \det(xI - BA) \end{align*}

And there it is.