What are the singular values of a matrix multiplied by the identity matrix?

In summary, the conversation discusses finding the singular values of a real mxn matrix with m>=n, written as A. The singular values are denoted by \sigma_j and are equal to \sqrt{1+\sigma_j^2} for the matrix (\stackrel{I_n}{A}). The conversation also mentions using an SVD to find the singular values and figuring out how to properly typeset the equation.
  • #1
azdang
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Homework Statement


Let A be a real mxn matrix, m>=n, with singular values [tex]\sigma[/tex]j.Show that the singular values of ([tex]\stackrel{I_{n}}{A}[/tex]) are equal to [tex]\sqrt{1+\sigma_j^2}[/tex].


Homework Equations





The Attempt at a Solution


I know that an SVD for A is A = U([tex]\stackrel{\Sigma}{0}[/tex])v^T and so, the singular values of A are [tex]\sigma_j[/tex]. I have no idea how to break this down. I assume I want to look at an SVD for ([tex]\stackrel{I_n}{A}[/tex]), but I don't know how to figure out that the singular values would be [tex]\sqrt{1+\sigma_j^2}[/tex]. Does anyone have any ideas? Thanks so much.
 
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  • #2
Also, sorry, I'm having a hard time figuring out how to have it typeset correctly to show you guys what's going on.
 
  • #3
Don't use the html tags and inside LaTex. Use _ for subscripts and ^ for superscripts.
 
  • #4
Oh wow, thank you so much. It looks great.
 

1. What is Singular Value Decomposition (SVD)?

Singular Value Decomposition (SVD) is a mathematical technique used to decompose a matrix into three matrices, representing the singular values, left singular vectors, and right singular vectors. It is commonly used in data analysis and machine learning for dimensionality reduction, data compression, and data approximation.

2. How does SVD work?

SVD works by breaking down a matrix into three components: U, Σ, and V. U represents the left singular vectors, Σ represents the singular values, and V represents the right singular vectors. The singular values are arranged in descending order and the corresponding singular vectors are aligned in the same order. This decomposition allows for the reduction of the dimensionality of the original matrix while preserving the most important information.

3. What are the applications of SVD?

SVD has various applications in fields such as data analysis, image processing, natural language processing, and recommender systems. It is used for dimensionality reduction, data compression, and data approximation, which are important techniques in machine learning and data mining. SVD is also used in signal processing, control systems, and other engineering applications.

4. What are the advantages of using SVD?

SVD has several advantages, including its ability to handle data with missing values, its robustness to noise, and its ability to capture the most important features of a dataset. It also allows for the reduction of dimensionality without losing much information, making it useful for data compression and data visualization. Additionally, SVD is a computationally efficient algorithm and can be easily implemented in various programming languages.

5. How is SVD different from other matrix decomposition techniques?

SVD is different from other matrix decomposition techniques, such as Eigendecomposition and QR decomposition, in that it can be applied to any type of matrix, including rectangular and non-square matrices. It also allows for the reduction of dimensionality while preserving the most important information, whereas other techniques may not have this capability. SVD is also more numerically stable and can handle matrices with repeated eigenvalues, making it a popular choice for many applications.

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