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Sylvester's determinant identity states that if $A$ and $B$ are matrices of sizes $m\times n$ and $n\times m$, then

$$ \det(I_m+AB) = \det(I_n+BA)$$

where $I_m$ and $I_n$ denote the $m \times m$ and $n \times n$ identity matrices, respectively.

Could you sketch a proof for me, or point to an accessible reference?

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4 Answers 4

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Hint $ $ Work universally, i.e. consider the matrix entries as indeterminates $\:\!\rm a_{\:\!ij},b_{\:\!ij}.\,$ Adjoin them to $\,\Bbb Z\,$ to get the polynomial ring $\rm R = \mathbb Z[a_{\:\!ij},b_{\:\!ij}].\, $ In this polynomial ring $\rm R,$ compute the determinant of $\rm\, (1+A B) A = A (1+BA)\,$ then cancel the nonzero polynomial $\rm\, det(A)\, $ (valid by $\rm R$ a domain). $ $ Extend to non-square matrices by padding appropriately with $0$'s and $1$'s to get square matrices. Note that the proof is purely algebraic - it does not require any topological notions (e.g. density).


Alternatively we may employ Schur decomposition as follows

$$\rm\left[ \begin{array}{ccc} 1 & \rm A \\ \rm B & 1 \end{array} \right]\, =\, \left[ \begin{array}{ccc} 1 & \rm 0 \\ \rm B & 1 \end{array} \right]\ \left[ \begin{array}{ccc} 1 & \rm 0 \\ \rm 0 & \rm 1\!-\!BA \end{array} \right]\ \left[ \begin{array}{ccc} 1 & \rm A \\ \rm 0 & 1 \end{array} \right]\qquad$$

$$\rm\phantom{\left[ \begin{array}{ccc} 1 & \rm B \\ \rm A & 1 \end{array} \right]}\, =\, \left[ \begin{array}{ccc} 1 & \rm A \\ \rm 0 & 1 \end{array} \right]\ \left[ \begin{array}{ccc} \rm 1\!-\!AB & \rm 0 \\ \rm 0 & \rm 1 \end{array} \right]\ \left[ \begin{array}{ccc} 1 & \rm 0 \\ \rm B & 1 \end{array} \right]\qquad$$


See this answer for more on universality of polynomial identities, universal cancellation (before evaluation) and closely relation topics, and see also this sci.math thread on 9 Nov 2007.

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    $\begingroup$ "simply pad-up appropriately with 0's and 1's to get square matrices." Oh, I can't believe it! Very nice! Many thanks. $\endgroup$ Jan 17, 2011 at 7:37
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    $\begingroup$ There is rarely need for anything... For example, there is no need for proofs to be purely algebraic :) $\endgroup$ Jan 17, 2011 at 7:42
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    $\begingroup$ @Mariano: It's a shining example of the power of universal proofs - which deserves emphasis (esp. since this simple algebraic proof is often overlooked - even by some professional mathematicians). $\endgroup$ Jan 17, 2011 at 15:31
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    $\begingroup$ "Schur decomposition" typically refers to Schur upper triangularization; it might be clearer to say that you are employing "the Schur complement". $\endgroup$ Feb 6, 2022 at 15:23
  • $\begingroup$ @BruceGeorge How to extend non-square matrices by padding with 0 and 1? $\endgroup$
    – log2
    Nov 10, 2023 at 10:08
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(1) Start, for fun, with a silly proof for square matrices:

If $A$ is invertible, then $$ \det(I+AB)=\det A^{-1}\cdot\det(I+AB)\cdot\det A=\det(A^{-1}\cdot(I+AB)\cdot A)=\det(I+BA). $$ Now, in general, both $\det(I+AB)$ and $\det(I+BA)$ are continuous functions of $A$, and equal on the dense set where $A$ is invertible, so they are everywhere equal.

(1) Now, more seriously:

$$ \det\begin{pmatrix}I&-B\\\\A&I\end{pmatrix} \det\begin{pmatrix}I&B\\\\0&I\end{pmatrix} =\det\begin{pmatrix}I&-B\\\\A&I\end{pmatrix}\begin{pmatrix}I&B\\\\0&I\end{pmatrix} =\det\begin{pmatrix}I&0\\\\A&AB+I\end{pmatrix} =\det(I+AB) $$

and

$$ \det\begin{pmatrix}I&B\\\\0&I\end{pmatrix} \det\begin{pmatrix}I&-B\\\\A&I\end{pmatrix} =\det\begin{pmatrix}I&B\\\\0&I\end{pmatrix} \begin{pmatrix}I&-B\\\\A&I\end{pmatrix} =\det\begin{pmatrix}I+BA&0\\\\A&I\end{pmatrix} =\det(I+BA) $$

Since the leftmost members of these two equalities are equal, we get the equality you want.

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    $\begingroup$ Nice argument. I guess this is very close to the "Schur decomposition" method suggested by Professor Dubuque. $\endgroup$ Jan 17, 2011 at 7:44
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    $\begingroup$ @Bruce, for the non-square situation you can argue similarly to the first part by using the fact that surjections $\mathbb R^n\to\mathbb R^m$, when $n\geq m$, are dense in the set of all matrices. $\endgroup$ Jan 17, 2011 at 7:46
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    $\begingroup$ @MarianoSuárez-Álvarez: Are you sure about the surjection argument? We still can't make sense of $\det A$ when $A$ is not square. $\endgroup$ Nov 19, 2018 at 8:03
  • $\begingroup$ @MarianoSuárez-Álvarez I can't see the equality, could u explain? $\endgroup$ Dec 12, 2022 at 14:10
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We will calculate $\det\begin{pmatrix} I_m & -A \\ B & I_n \end{pmatrix}$ in two different ways. We have $$ \det\begin{pmatrix} I_m & -A \\ B & I_n \end{pmatrix} = \det\begin{pmatrix} I_m & 0 \\ B & I_n + BA \end{pmatrix} = \det(I_n + BA). $$ On the other hand, $$ \det\begin{pmatrix} I_m & -A \\ B & I_n \end{pmatrix} = \det\begin{pmatrix} I_m+AB & 0 \\ B & I_n \end{pmatrix} = \det(I_m + AB). $$

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    $\begingroup$ Maybe it's useful to show why you can do like that, that is, by multiplying on the right or on the left by the determinant $1$ matrix $\begin{pmatrix}I_m & A\\ 0 & I_n\end{pmatrix}$. $\endgroup$
    – egreg
    Jan 5, 2017 at 11:46
  • $\begingroup$ What exactly is that determinant? Its elements are matrices itself? $\endgroup$ Jan 5, 2017 at 11:50
  • $\begingroup$ As I mentioned in my post in the lnked dupe, this is a special case of Schur decomposition. $\endgroup$ Jan 5, 2017 at 14:45
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here is another proof of $det(1 + AB) = det(1+BA).$ We will use the fact that the nonzero eigen values of $AB$ and $BA$ are the same and the determinant of a matrix is product of its eigenvalues. Take an eigenvalue $\lambda \neq 0$ of $AB$ and the coresponding eigenvector $x \neq 0.$ It is claimed that $y = Bx$ is an eigenvector of $BA$ corresponding to the same eignevalue $\lambda.$
For $ABx = Ay = \lambda x \neq 0,$ therefore $y \neq 0.$ Now we compute $BAy = B(ABx) = B(\lambda x) = \lambda y.$ We are done with the proof.

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    $\begingroup$ You need to show further that a non-zero eigenvalue has the same algebraic multiplicity for both $AB$ and $BA$. Only after that can you conclude the proof. $\endgroup$
    – me10240
    Sep 19, 2019 at 16:02

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