What are the requirements for SVD to work?

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Discussion Overview

The discussion revolves around the requirements for Singular Value Decomposition (SVD) to work, particularly in the context of calculating pseudoinverses. Participants explore the implications of matrix properties, such as eigenvalues and independence of vectors, on the ability to compute pseudoinverses and the applicability of SVD.

Discussion Character

  • Exploratory
  • Technical explanation
  • Debate/contested
  • Mathematical reasoning

Main Points Raised

  • One participant notes that the matrix under consideration has a zero eigenvalue, questioning whether non-zero eigenvalues are necessary for finding a pseudoinverse.
  • Another participant references the Moore-Penrose pseudoinverse and its defining properties, seeking clarification on its application in the context of the problem.
  • A participant suggests that the last system of equations may not have a solution due to the presence of dependent column vectors in the matrix.
  • There is a mention of using a least squares approach as a potential method to check solutions, although the specifics of this approach are not fully explored.
  • One participant expresses uncertainty about the requirements for SVD, initially stating that it requires positive definiteness, but later corrects themselves to clarify that this requirement applies to Cholesky Decomposition instead.
  • Another participant highlights that SVD is sensitive to variations in the values of the matrix, indicating that small numbers can significantly affect the results.

Areas of Agreement / Disagreement

Participants do not reach a consensus on the specific requirements for SVD or the implications of matrix properties on the ability to compute pseudoinverses. Multiple competing views and uncertainties remain regarding the relationship between matrix characteristics and the applicability of SVD.

Contextual Notes

There are unresolved assumptions regarding the definitions of matrix properties and the implications of eigenvalues on the existence of solutions. The discussion also reflects a lack of clarity about the distinctions between different matrix decomposition methods.

spookyfw
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Dear fellows,

during my internship I've stumbled over a problem of analysis. To cut things short some pseudoinverses have to be calculated. For one of them it does not work, s.t. A'*A [itex]\neq[/itex] I.

I just wondered about the requirements to find a pseudoinverse. One of the eigenvalues is zero, but as far as I understand that eigenvalues different than zero are not a requirement.

The matrix under consideration is that one:

A = [1 -1 0; 1 0 -1; 0 -1 1; 0 -1 1];

If anyone could help me that would be really great :).

Cheers and thank you very much in advance,
spookyfw
 
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Hey spookyfw and welcome to the forums.

I just took a look at the Moore-Penrose pseudo-inverse from a grad linear algebra book and for that pseudo-inverse, the relation ship is that BAB = A and ABA = B.

Are you using this one or some other one?
 
Hi chiro,

thank you very much for your reply. Yes I am using the Moore-Penrose pseudo-inverse. Actually the original problem can be seen in the attached picture. After using the pseudo-inverses it is stated that only the first three systems have a solution, but that there is too little information to solve for A00, B00 and C00. When I am multiplying with the pseudo-inverses I get the unitary matrix on the left hand side, but for the last system. So I wondered how that is related to the last one not having a solution and what the reasoning is. I thought about independence, but there is also row-degeneracy in the matrix of the first system.

I hope that this still makes sense. Any idea?
spookyfw
 

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I'm wondering just as a curiosity, whether you have tried using a least squares approach to get some kind of reference for your solutions?
 
hmm..what do you mean by that least square approach? I just wanted to check a solution, I just don't understand the mathematical reason why the last system cannot be solved for A00, B00 and C00.
How would you go about the least square approach though?
 
Thank you very much. I think I spotted the problem with the matrix, as for square-matrices the Moore-Penrose inverse needs independent vectors, but unfortunately there is two dependent column vectors. At first sight I just spotted degenerate row vectors, but of course there has to be dependent ones with a 4x3.
 
Hi spookyfw,

I'm a bit rusty on this now, but I believe SVD requires the matrix to be positive definite. That's probably true for any procedure based on Spectral Decomposition.
 
Sorry, it's Cholesky Decomposition that requires the matrix to be positive definite. In that case, Cholesky Decomposition is far quicker to arrive at a result. The SVD is especially sensitive to a large variation in the range of cells values. Very small numbers can wreak havoc with the results.
 

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