Coninvolution: A Powerful Property of Nonsingular Matrices

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Discussion Overview

The discussion centers around the concept of coninvolutory matrices, specifically focusing on the properties of nonsingular matrices in the context of the lemma that states if \( A \) is a nonsingular matrix, then \( \overline{A}^{-1}A \) is coninvolutory. Participants seek clarification, examples, and literature related to this topic.

Discussion Character

  • Exploratory
  • Technical explanation
  • Debate/contested
  • Homework-related

Main Points Raised

  • One participant defines a coninvolutory matrix as one where \( A^{-1} = \overline{A} \) and seeks to prove a lemma regarding nonsingular matrices.
  • Another participant suggests that the lemma can be proven by manipulating the expression \( \left(\left(\overline{A}\right)^{-1}A\right)^{-1} \) and using matrix identities.
  • Participants discuss the process of finding examples of coninvolutory matrices by calculating \( \left(\overline{A}\right)^{-1}A \) for various nonsingular matrices.
  • There is a suggestion that a matrix \( A \) can be expressed as \( A = B + i C \) for real matrices \( B \) and \( C \), leading to conditions for coninvolutory matrices.
  • Clarification is sought regarding the conditions \( A^2 + B^2 = I \) and \( AB = BA \), with a later confirmation that the conditions should indeed involve \( B^2 + C^2 = I \) and \( BC = CB \).

Areas of Agreement / Disagreement

Participants express varying degrees of understanding and clarity regarding the lemma and examples of coninvolutory matrices. While some participants provide helpful insights and clarifications, there remains uncertainty about the terminology and the specific properties of these matrices.

Contextual Notes

There is a lack of established literature on coninvolutory matrices, and some participants express difficulty in finding clear definitions or examples. The discussion also reveals potential confusion regarding the conditions for coninvolutory matrices when expressed in terms of real and imaginary components.

Who May Find This Useful

This discussion may be useful for those interested in advanced linear algebra, particularly in the study of matrix properties and their applications in complex analysis or related fields.

BrainHurts
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It's very difficult for me to find any simple literature to explain this idea.

J[itex]\in[/itex]Mn(ℂ) is a coninvolutory (or a "coninvolution") if A-1=[itex]\overline{A}[/itex]
I'm looking to prove this lemma:

Let A be an element of Mn(ℂ) and A is nonsingular, then [itex]\bar{A}[/itex]-1A is coninvolutory.

I see that the identity matrix is a coninvolution. Does anyone have another example?
 
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It is unclear what you are asking for. Are you asking
1) How to prove your lemma,
2) for examples of coinvolutory matrices, or
3) for general litterature on coinvolutory matrices?

The lemma is fairly straightforward to prove by calculating [itex]\left(\left(\overline{A}\right)^{-1}A\right)^{-1}[/itex] and showing that it is equal to [itex]\overline{\left(\overline{A}\right)^{-1}A}[/itex] using identities like
[tex](AB)^{-1}=B^{-1}A^{-1} \qquad \overline{AB} = \overline{A}\,\overline{B}[/tex]
If you are having trouble proving it, then tell us where you get stuck.

If you need examples of coinvolutory matrices, then just pick some nonsingular matrix A and calculate [itex]\left(\overline{A}\right)^{-1}A[/itex] as the lemma suggests. Alternatively if you write A = B + i C for real matrices B and C, then you can show that A is coinvolutory precisely if A^2 + B^2 = I and AB=BA. In particular if A is a real matrix, then it is coinvolutory if and only if it is involutory (i.e. iff A^2=I).

If you want to read more about coinvolutory matrices, then I can't help you as I have never heard of the term and a quick google search does not reveal much.
 
I was asking for help on all three and your post really helped a lot! I'll try to be more clear in the the near future. Thank so much!
 
rasmhop said:
If you need examples of coinvolutory matrices, then just pick some nonsingular matrix A and calculate [itex]\left(\overline{A}\right)^{-1}A[/itex] as the lemma suggests. Alternatively if you write A = B + i C for real matrices B and C, then you can show that A is coinvolutory precisely if A^2 + B^2 = I and AB=BA. In particular if A is a real matrix, then it is coinvolutory if and only if it is involutory (i.e. iff A^2=I).

Do you mean B^2 +C^2 = I and BC=CB?
 
BrainHurts said:
Do you mean B^2 +C^2 = I and BC=CB?

Yes.
 

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