Negative Value Matrix: Finding the Singular Value Decomposition

In summary, the goal of the problem is to find the singular value decompositions (SVD) of a matrix. Once we do this, it will be comprised of three different matrices. The first matrix is the eigenvector of the matrix A'A. The second matrix is the eigenvector of AA'. The third matrix is the eigenvector of A.
  • #1
asif zaidi
56
0
I did some problems from the example and the questions at end of chapter. I got all of them right except this one.

Problem Statement:

Consider the matrix [3 0; 0 -2]. Find its singular value decompositions

Problem Solution

Goal is to find A = U*S*V as below

Step1: Find AA', A'A. In this case they both are equal and are [9 0; 0 4];

Step2
: Find U = eig vector (AA'). Doing so gives [1 0; 0 1];

Step 3: Find S = [3 0; 0 2] (I am not showing the steps)

Step 4: Find V = eig vector (A'A). Doing so gives [1 0; 0 1];

Verify: Multiply U*S*V and it should give back A.

My problem is it gives [3 0; 0 2] which is different than A = [3 0; 0 -2].

I know that if I change V to [1 0 ; 0 -1] I will get A back. But why do my computations not show this. What am I missing?

Like I said, I did the above procedure for a lot of other numbers and I get it right. Only when I have a negative value in the matrix then it seems I am missing a -1 factor which I cannot get from my procedure.

Thanks

Asif
 
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  • #2
I'm not really sure exactly what recipe you are following. Can you tell us where you found the algorithm? U could also be [[1,0],[0,-1]]. I'm not sure what 'eig vector(AA')' means, for example.
 
  • #3
Hi:

I am following the recipe from Linear Algebra with Applications (Steven J Leon)

Basically the text says on pg 346

- V matrix is the eigenvector of A'A
- U matrix is the eigenvector of AA'

I agree with you that U could also be as you said... But why didn't my calculations come up with that.

Thanks

Asif
 
  • #4
Well, I asked for that. I don't have that reference so I can't really tell you. Maybe somebody else does. But I still don't get exactly what "V matrix is the eigenvector of A'A" means. [0,1] is an eigenvector of A'A. But so is [0,-1]. I don't know how you are supposed to put the signs in. Maybe just by hand?
 
Last edited:
  • #5
I am not typing the whole thing, so perhaps it is creating confusion

- The goal is to find SVD of a matrix A. Once we do this it will be comprised of 3 different matrices, U, S, V

- The text explains how to get S. It says compute A*A' and take the square root of the eigenvalues

- To get u: compute A*A' and find the eigen vector.

The example which I have, one can easily see that (1,1) or (1,-1) is the eigenvector. But what if I am given a problem in an exam which I cannot easily see. This is my main concern. I would have thought the math computation would have given this result.

Hope this clarifiesThanks

Asif
 

What is a negative value matrix?

A negative value matrix is a type of matrix where at least one of the elements is a negative number. This means that the matrix contains both positive and negative values.

What is the purpose of finding the singular value decomposition of a negative value matrix?

The singular value decomposition (SVD) of a matrix is a powerful tool used to break down a matrix into simpler components. In the case of a negative value matrix, the SVD can help identify the underlying patterns and relationships within the data, which can be useful in various applications such as data compression and data analysis.

How is the singular value decomposition of a negative value matrix calculated?

The singular value decomposition of a matrix involves finding the eigenvalues and eigenvectors of the matrix. This can be done using various numerical methods, such as the power method or the Jacobi method. Once the eigenvalues and eigenvectors are found, the SVD can be calculated using a specific formula.

What is the difference between the singular value decomposition of a negative value matrix and a positive value matrix?

The main difference is that the singular values (diagonal elements of the SVD) of a negative value matrix will be a combination of positive and negative values, whereas in a positive value matrix, all singular values will be positive. Additionally, the eigenvectors of a negative value matrix will also be a combination of positive and negative values.

What are some practical applications of the singular value decomposition of a negative value matrix?

The SVD of a negative value matrix can be applied in various fields, such as image and signal processing, data compression, and recommendation systems. It can also be used in statistics and data analysis to identify the underlying patterns and relationships within the data.

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