Determinant of a non-square matrix?

[GargleBlast42]

"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.

 

[Lord Crc]

There's probably a better solution, but from what I can see, if the rank is n, then it should be http://en.wikipedia.org/wiki/Inverse_element#Matrices", and thus AA^T should exist and be invertible. So the determinant of AA^T could perhaps fit the bill?

 

[NaturePaper]

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

 

[Landau]

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

Regards

Singular values.

 

[GargleBlast42]

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?

 

[trambolin]

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