SVD when the V matrix is zero?

In summary: But in this case, there are no repeated singular values so U and V should be unique.In summary, the conversation discusses a problem with finding the singular value decomposition (SVD) of a matrix. The question asks for an explanation about one of the U or V matrices turning out to be zero, but the answer key only provides the general formula for SVD. The conversation also discusses the uniqueness of the SVD and the possibility of U and V being zero. It is suggested to find the eigenvalues and eigenvectors of the matrix to solve the problem.
  • #1
SELFMADE
80
0
Semester is over but still want to figure this out

Prob 8 on here:

http://www.math.uic.edu/~akers/310PracticeFinal.pdf

When trying to SVD that matrix one of the U or V matrix turns out to be zero but the answer key has just the general formula for SVD

can anyone explain thanks
 
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  • #2
SELFMADE said:
Semester is over but still want to figure this out

Prob 8 on here:

http://www.math.uic.edu/~akers/310PracticeFinal.pdf

When trying to SVD that matrix one of the U or V matrix turns out to be zero but the answer key has just the general formula for SVD

can anyone explain thanks

U and V can't be zero. The the defintion of the SVD says they are orthonormal matrices.

How are you doing this by hand? One way is to finding the eigenvalues and vectors of A^T.A and A.A^T . The U and V matrices are the eigenvectors of those matrices, so it doesn't make any sense for them to be zero.
 
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  • #3
Since the eigenvalues of O*O are all 0, Σ would be the same exact zero matrix as the given. SVD is unique only to Σ in general, so you can pick any orthonormal bases for your domain and codomain and you have U and V.
 
  • #4
zcd said:
SVD is unique only to Σ in general, so you can pick any orthonormal bases for your domain and codomain and you have U and V.

Are you sure about that?

A = U S VT

AT A = V S UT U S VT = V S2 VT

AT A V = V S2 VT V = V S2

So S2 and V are the eigenvalues and vectors of AT A

Except in the special case where there are repeated eigenvalues, V is uniquely defined (apart from scaling factors of +/-1).

Starting with A AT, the same is true of U.
 
  • #5
But in the case of T:R^2 -> R^2 where T is the zero transformation, any two linearly independent vectors from R^2 can be eigenvectors of T. Even with the restriction that the two eigenvectors we pick from R^2 are orthogonal, there's more than one way we can pick the vectors aside from ordering. Since his matrix in question is specifically the zero matrix, the pathological example is actually relevant.
 
  • #6
I think the OP's problem was that he got a zero V matrix, but that is wrong. Since we don't know what he/she actually did to get V = 0, it's impossible to say what the mistake was.

If you look at the question in the OP's link, the matrix does not have any repeated singular values.

I agree that U and V are not unique if there are repeated singular values.
 

1. What is SVD when the V matrix is zero?

SVD (Singular Value Decomposition) is a mathematical technique used to decompose a matrix into smaller, simpler matrices. When the V matrix is zero, it means that the original matrix can be represented by only two matrices, U and D. This is known as the reduced SVD or economy SVD.

2. Why is the V matrix sometimes zero in SVD?

The V matrix can be zero in SVD when the original matrix has a special structure, such as being symmetric or having a high degree of correlation between its columns. In these cases, the information in the original matrix can be fully represented by the U and D matrices, making the V matrix redundant.

3. How does the V matrix being zero affect the computation of SVD?

When the V matrix is zero, the computation of SVD becomes simpler and more efficient. This is because the V matrix does not need to be calculated, reducing the number of computations needed. As a result, reduced SVD is often preferred over the full SVD for large matrices.

4. Can the V matrix be zero in all cases?

No, the V matrix cannot be zero in all cases. It is only zero when the original matrix has a special structure, as mentioned earlier. If the original matrix does not have this structure, then the V matrix will not be zero and the full SVD will be needed to decompose the matrix.

5. What are the applications of SVD when the V matrix is zero?

The reduced SVD has various applications in data analysis, signal processing, and machine learning. It is commonly used for dimensionality reduction, data compression, and feature extraction. It is also used in collaborative filtering, where it helps to make recommendations based on user-item ratings.

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