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The definition of the adjugate matrix is easy to understand, but I have never seen it used for anything.

  • What is the intuitive meaning of this matrix?

  • Are there examples of applications which may shed light on its conceptual meaning? I would be especially interested to hear examples of usage in representation theory or other abstract algebra disciplines.

Alexander Gruber
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1 Answers1

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You have never seen it used for anything?

The fact that $A\text{adj}(A)=\text{adj}(A)A=\det(A)I$ is the standard one-half of the proof that $A\in\text{GL}_n(R)$ iff $\det(A)\in R^\times$, where $R^\times$ is the group of units of $R$.

The conceptual meaning of the adjugate matrix is somewhat complicated. Really, you can imagine it as being the adjoint of $\bigwedge^{n-1} A$ with respect to a somewhat natural pairing. More information can be found here.

Mike Pierce
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Alex Youcis
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  • It's the natural pairing, not just a somewhat natural pairing. If you do the same thing with the symmetric algebra, use the permanent instead of the determinant, and look at adjoints of multiplication, you immediately reinvent partial derivatives. – Joshua P. Swanson Sep 21 '19 at 03:27
  • @JoshuaP.Swanson I am sorry for replying to a 4 years old comment, but do you mind giving some reference book or article where I can read more about what you said here? I do have some experience with using the determinant and the adjugate matrix (or, more precisely, the cofactor matrix) in analysis and have some basic knowledge of multilinear algebra, so I'd like to understand more about the viewpoint you mentioned. – BigbearZzz Jun 13 '23 at 19:06
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    @BigbearZzz I have a paper with Nolan Wallach that explicitly defined the natural prefect pairings for exterior and symmetric algebras and connects the adjoints of multiplication to differential operators. See Sections 2.1-2.3. The material was well-known, but hard to cite. (You can ignore the G.) – Joshua P. Swanson Jun 13 '23 at 23:26
  • @JoshuaP.Swanson Very interesting, thank you. Do you have any recommendation regarding a textbook/article that discusses the uses of multilinear algebra in analysis? Over the years, I have seen the topic discussed here and there, but I don't know a good resource that is somewhat at an introductory (graduate) level. From what I've seen, it is usually included as a complimentary background or in the appendix of a paper. – BigbearZzz Jun 14 '23 at 19:22
  • A recent book "Geometric Multivector Analysis" by Rosén is similar to what I am looking for, but a more analysis-oriented book would be preferable. – BigbearZzz Jun 14 '23 at 19:26
  • I'm probably not the right person to ask, but I suppose any book that discusses de Rham cohomology would count, since differential forms are built from multilinear algebra. Jack Lee's smooth manifolds book is well-regarded, though it's probably more algebraic than you have in mind. – Joshua P. Swanson Jun 14 '23 at 23:36