Determinant of a non-square matrix?

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Discussion Overview

The discussion revolves around the search for a numerical invariant that characterizes the rank of a non-square matrix, akin to the determinant for square matrices. Participants explore potential methods to determine conditions under which a non-square matrix does not achieve full rank, without resorting to Gaussian elimination.

Discussion Character

  • Exploratory, Technical explanation, Debate/contested

Main Points Raised

  • One participant inquires about a numerical invariant for non-square matrices that would indicate rank, suggesting a relationship to determinants.
  • Another participant proposes that the determinant of the product AA^T could serve as a suitable invariant if the rank is n.
  • A later reply questions the terminology of eigenvalues related to AA^\dagger, identifying them as singular values.
  • One participant shares their experience with Mathematica's limitations in handling symbolic matrices with parameters, particularly when dealing with the determinant equation involving complex conjugates.
  • Another participant suggests using QR decomposition or solving linear equations as alternative methods to explore the rank conditions of the matrix.

Areas of Agreement / Disagreement

Participants express various approaches and ideas, but there is no consensus on a definitive method or solution for characterizing the rank of non-square matrices.

Contextual Notes

Limitations include the dependence on specific matrix forms and the challenges posed by complex conjugates in determinant equations, which may affect the applicability of proposed methods.

GargleBlast42
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"Determinant" of a non-square matrix?

Hi,

is there any numerical invariant that would characterize the rank of a non-square matrix, similar to the determinant for square matrices? I.e. having a matrix nxm, with n<m, I'm looking for a number that would be zero if the rank of the matrix is smaller than n and nonzero if the rank is n. By "similar to the determinant" I mean that it would be some number, which you could obtain by doing some arithmetic operations on the entries, but without the necessity to perform Gaussian Elimination.
 
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Lord Crc said:
So the determinant of AA^T could perhaps fit the bill?

I hope you are right...by the way what the eigenvalues of AA^\daggers called? :)

Regards
 


NaturePaper said:
I hope you are right...by the way what the eigenvalues of AA^\daggers called? :)

Regards
Singular values.
 


Thanks Lord Crc, I have actually thought about this before, but it turns out that Mathematica can't handle this very well for large symbolic matrices.

Maybe a bit more about my problem: my matrix has one parameter and I want to find out for which values of this parameter this matrix doesn't have its full rank. Computing the determinant of the matrix isn't a problem for Mathematica, but the equation Det[A A^\dagger]=0 which I obtain contains complex conjugates and it seems that Mathematica is not able to deal with this kind of equation.

Finding singular values seems also uneffective. Any other ideas?
 


QR decomposition might be fast. And solving linear equations maybe? Example:
<br /> \begin{pmatrix}a &amp;b\\c&amp;d\\e&amp;f\\g&amp;h<br /> \end{pmatrix}x = 0<br />
gives you relations about the entries. You can check the roots of the resulting possible polynomials.
 

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