How is a non-quadratic matrix singular?

[divB]

Hi,

I have a matrix A (in general not quadratic). It has an arbitrary amount of rows and L+1 columns. This matrix should be of rank K.

But using the MATLAB command

rank(K)

often gives the wrong result.

However, according to a paper this is another approach to identify K. It says: "Find the smallest L such that the matrix A is singular. Then K = L − 1."

But: A is not quadratic in general. How to I see if a non-quadratic matrix is singular? What would be the appropriate way to do this in MATLAB?

I do not understand this: I can not calculate the determinant of a non-quadratic matrix...

Regards,
divB

 

[Office_Shredder]

What is a quadratic matrix? Do you mean a square matrix?

Why do you have rank(K) as a MATLAB command? Isn't K a number that's supposed to be the rank of A?

For your discussion about L... how do you get to pick L? If L is different you have a different matrix. The technique doesn't make any sense as you've written it.

As to determining whether a matrix is singular... if it doesn't have the same number of rows and columns, the term makes no sense. Matrices that don't have the same number of rows and columns cannot be invertible (because the domain and the codomain don't have the same dimension, so there can't be a linear bijection between them). Singular is usually reserved to talk about square matrices that are not invertible; I suppose if you want to apply the term to non-square matrices all of them would be singular.

 

[divB]

Hi,

Thank you very much for your reply!

First: Yes of course, I mean square matrix!

Second: Because the rank K tells me the number of linear independent rows/columns in A. In fact the matrix A is built using a specific algorithm (which I will describe shortly) and I know from this fact what *should* be the rank.

Third: I know that the matrix will be different when I change L. However, as said, the matrix is built up by a specific algorithm. This way it should be possible to determine the rank with this method. More about it: later.

And to your last statement: Are you sure? This way the paper would be wrong ;-)

However, maybe it is easier to understand if I just write how the matrix is built.

First I have K values t_k. In my case they are known a priori but in real they are not. Also, the amount is not known in reality.

From these t_k I build sequences s_m as follows:

$$s_m = \sum_{k=1}^K c_m t_k$$

where the c_m are just constants (known a priori). The number of sequences is N where N = 2*L - 1 >> 2*K.

Using this sequence, the following Toeplitz matrix A is built:

for l=1:N-L+1
    A(l,:) = tau(L+l:-1:l);
end

I wrote MATLAB code because it is easier to write. I hope it is possible to understand.

As you can see, this matrix has L columns now and a specific amount of rows (N-L+1) but this is not important.

Important is the following fact: The matrix is not square but because of its construction the rank must not exceed K! Because the sequences s_m just consists of a linear combination of K distinct values.

The big question is: How to find K? (As said before: In reality the matrix A is not built that way and therefore I do not know K)

My naive way was: rank(A). This works for small values of K but for large values the result is always wrong.

The second approach is from the paper and is exactly what I quoted in the first post: "Find the smallest L such that the matrix A is singular. Then K = L − 1."

This works in theory because L >> K.

I hope my setup is clear now :-)

Regards,
divB

 

[Dickfore]

There is no determinant of a matrix that is not square.

 

[divB]

Hi,

Yes I know that there is no determinat ... but is there a way to tell if a matrix is just singular?

Or is the paper wrong?

http://preview.tinyurl.com/3yf56kx

Page 7 under equation (13) you can see exactly what I have written in my post. It clearly states that a non-rectangular matrix can be singular/non-singular.

Regards,
divB

 

[Dickfore]

please quote the exact paragraph where it gives a criterion for a rectangular matrix being singular/non-singular, cause I can't find it.

 

[divB]

Hi,

Looking at equation (12) (page 7) you can see that A is rectangular (not square) because it has 2M-L+1 rows and L+1 columns. Furthermore, the paper states directly after this definition:

[...] which can be seen as a rectangular extension of (8)

And directly afterwards you can find:

[...] Find the smallest L such that the matrix A built according to (12) is singular. Then K = L − 1.

So the paper states that there must exist a definition for singularity of a rectangular matrix.

Regards,
divB

 

[Office_Shredder]

Yes when it says that a rectangular matrix can be singular, it's wrong. If L>K they describe how the matrix has a non-trivial kernel, so I suspect what the author intends is to say find the smallest value of L such that there is a non-trivial kernel, or something to that effect

 

[Dickfore]

Oh, I see.

Do you know when the homogeneous system:

$$\mathbf{A} \cdot H = 0$$

with $\mathbf{A}$ being a rectangular matrix, has a non-trivial solution, i.e. a vector $H$ that is not identically equal to zero?

 

[divB]

Hi,

Thank you for the hint. Ok, this maybe :-) Unfortunately I am not mathematican ;-)

Do you have any hint of this topic? How can I see if there exists a (non)trivial kernel for a matrix?

Maybe a question related to this: I explained why the matrix A must have a rank of K. What could be the reason why MATLAB gives wrong results of the "real" K becomes larger?

Is there a different way to calculate/estimate the rank of a matrix?

What I tried was to calulcate the condition number of the matrix A in relation to L, i.e. "cond(A(L))". But I could not find a significant "step" at L=K...

Regards,
divB

 

[divB]

@Dickforce: Yes, I think so: If A has full rank? In terms of my example: If L=K ?

 

[Dickfore]

The rank of a matrix is defined as the highest possible dimension of a (square) sub-determinant that is non-zero (non-singular). Let $\mathbf{A}_{m \times n}$ and let $k = \mathrm{rank}(\mathbf{A})$. The condition that a non-trivial solution exist is that $k < n$.

 

[divB]

Hi,

Hmm. I do not fully understand what you mean. In fact it seems clear. But what if do not know $$k = \mathrm{rank}(\mathbf{A})$$ and I want to find it?

Regards, divB

 

[Dickfore]

I simply stated the condition for a non-trivial solution to exist. I don't say anything about finding the rank.

 

[divB]

Oh, ok that's clear. My question was how to find the rank..
Thanks anyway!

 

[Dickfore]

There are many standard methods in the literature. You should read up on them.

 

[divB]

And the method proposed in the paper ("Find the smallest L such that the matrix A built according to (12) is singular. Then K = L − 1") is wrong in your opinion?

Regards,
divB

 

[Dickfore]

I don't know. I don't have time to read through the paper. Sorry.

 

[divB]

Ok, thanks anyway!