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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?

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    $\begingroup$ 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... $\endgroup$ – Qiaochu Yuan Feb 22 '13 at 21:12
  • $\begingroup$ Is the eigen value argument another one way? Can I say that the eigen values of $AB$ and $BA$ are same so the characteristics equations are same?@QiaochuYuan $\endgroup$ – cmi Nov 19 '18 at 4:56
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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.

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    $\begingroup$ 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? $\endgroup$ – EuYu Feb 22 '13 at 17:37
  • $\begingroup$ @EuYu, I have no reference, sorry. $\endgroup$ – Nathan Portland Feb 22 '13 at 22:11
  • $\begingroup$ Is the eigen value argument another one way? Can I say that the eigen values of $AB$ and $BA$ are same so the characteristics equations are same? @NathanPortland $\endgroup$ – cmi Nov 19 '18 at 4:55
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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.

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  • $\begingroup$ Is the eigen value argument another one way? Can I say that the eigen values of $AB$ and $BA$ are same so the characteristics equations are same?@M.H $\endgroup$ – cmi Nov 19 '18 at 4:56
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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?

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    $\begingroup$ This shows that $AB$ and $BA$ have different minimal polynomial. But characteristic polynomials are the same right? $\endgroup$ – Unknown x Nov 25 '17 at 1:39
  • $\begingroup$ Is the eigen value argument another one way? Can I say that the eigen values of $AB$ and $BA$ are same so the characteristics equations are same?@Jim $\endgroup$ – cmi Nov 19 '18 at 4:56
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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 $(BA)^k=B(AB)^{k-1}A$.

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  • $\begingroup$ in general means for m not equal to n?? $\endgroup$ – user152715 Sep 16 '14 at 21:16
  • $\begingroup$ 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. $\endgroup$ – Marc van Leeuwen Dec 23 '15 at 21:10
  • $\begingroup$ Is the eigen value argument another one way? Can I say that the eigen values of $AB$ and $BA$ are same so the characteristics equations are same?@MarcvanLeeuwen $\endgroup$ – cmi Nov 19 '18 at 4:58
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For squre matrix, the charateristic polynomials are same, but for $A$ is matrix of size $m \times n$ and B be matrix of size $n \times m$. Then $x^{m}C_{BA}(x)=x^{n}C_{AB}(x)$ this implies that the nonzero eigenvalue of $AB$, counted with multiplicites, 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 eigenvalue 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.

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