How to Calculate the Pseudoinverse Using the SVD?

In summary: To change the letter S to S+ in the image, you would have to change the values of the small elements to something larger than epsilon.
  • #1
nao113
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Homework Statement
Hello, I have question related to SVD. Can anyone give me hints about what should I do to solve this question. I provided the question and my answer on the picture below. I have found the eigenvalues and eigenvector but I m not sure whether it is correct not and then whether this answer already cover the question. Thank you
Relevant Equations
𝐀=𝐔Σ𝐕∗
Screen Shot 2022-05-21 at 16.05.23.png


My Answer:
I am still beginner in this area so it s quite hard for me to understand this one. I am not sure what the output that this question asked me. I thought it might be asked about the value of x1, x2, x3, and x4

WhatsApp Image 2022-05-22 at 4.32.31 PM.jpeg
 
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  • #2
Recheck that determinant for ##~\mathbf{H^T H - I} \it \lambda ## .
 
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  • #3
SammyS said:
Recheck that determinant for ##~\mathbf{H^T H - I} \it \lambda ## .
Hello, thank you the respond, actually, I have correct my mistakes here. I already got the answer from this. But still, can you help me to understand what is U means? and how to get the matrix for it as well as Vt?
 

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  • #5
nao113 said:
Hello, thank you the respond, actually, I have correct my mistakes here. I already got the answer from this. But still, can you help me to understand what is U means? and how to get the matrix for it as well as Vt?
U is one of the parts of the SVD, which is given by U Σ VT.
See: https://en.wikipedia.org/wiki/Singular_value_decomposition
Sometimes the letter S is used in place of Σ, which is the case in the image you've linked to.

The image you linked is trying to show how to calculate the pseudoinverse using the SVD: https://en.wikipedia.org/wiki/Moore–Penrose_inverse. See the heading about 2/3 of the way down the page: "Singular value decomposition (SVD)"

but they have made a mistake. They show the pseudoinverse as: US+VT, but it should be VS+UT

The numeric values they show are correct.
 
  • #6
The Electrician said:
U is one of the parts of the SVD, which is given by U Σ VT.
See: https://en.wikipedia.org/wiki/Singular_value_decomposition
Sometimes the letter S is used in place of Σ, which is the case in the image you've linked to.

The image you linked is trying to show how to calculate the pseudoinverse using the SVD: https://en.wikipedia.org/wiki/Moore–Penrose_inverse. See the heading about 2/3 of the way down the page: "Singular value decomposition (SVD)"

but they have made a mistake. They show the pseudoinverse as: US+VT, but it should be VS+UT

The numeric values they show are correct.
do you know how to change S to S+?
 
  • #7
nao113 said:
do you know how to change S to S+?
Re-read the section I linked to:

The image you linked is trying to show how to calculate the pseudoinverse using the SVD: https://en.wikipedia.org/wiki/Moore–Penrose_inverse. See the heading about 2/3 of the way down the page: "Singular value decomposition (SVD)"

"For a rectangular diagonal matrix such as S, we get the pseudoinverse by taking the reciprocal of each non-zero element on the diagonal, leaving the zeros in place, and then transposing the matrix. In numerical computation, only elements larger than some small tolerance are taken to be nonzero, and the others are replaced by zeros." Any of the elements that are very small (comparable to to the value of epsilon in the arithmetic of the math system you're using) are replaced by zero, not reciprocated."
 
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Related to How to Calculate the Pseudoinverse Using the SVD?

What is Singular Value Decomposition?

Singular Value Decomposition (SVD) is a mathematical method used to decompose a matrix into three matrices, namely U, Σ, and V. It is used to reduce the dimensionality of a dataset and extract important features from the data.

Why is Singular Value Decomposition important?

Singular Value Decomposition is important because it helps in data compression, noise reduction, and feature extraction. It is also used in various applications such as image processing, data mining, and natural language processing.

What are the applications of Singular Value Decomposition?

Singular Value Decomposition has various applications in different fields. In mathematics, it is used for solving linear systems of equations and computing pseudoinverses. In statistics, it is used for principal component analysis. In machine learning, it is used for dimensionality reduction and feature extraction. Other applications include image and signal processing, data compression, and recommendation systems.

What is the difference between Singular Value Decomposition and Eigenvalue Decomposition?

The main difference between Singular Value Decomposition (SVD) and Eigenvalue Decomposition (EVD) is that SVD can be applied to any rectangular matrix, while EVD can only be applied to square matrices. Additionally, SVD produces three matrices (U, Σ, and V), while EVD produces only two matrices (eigenvectors and eigenvalues).

How is Singular Value Decomposition calculated?

Singular Value Decomposition is calculated using various algorithms such as the Jacobi method, the power method, and the QR algorithm. These algorithms use iterative techniques to compute the decomposition of a matrix. In addition, there are various software packages that can perform SVD calculations, such as MATLAB, Python's NumPy library, and R's svd function.

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