Proof of Singular Value Decomposition

In summary, the conversation discusses two parts: 1) proving the existence of unitary matrices in an arbitrary linear map of a complex vector space, and 2) showing that this holds if and only if there exist orthonormal bases that satisfy certain conditions. The speaker has successfully proven part 1, but is stuck on part 2 and is seeking help. They are unsure how to approach the proof in both directions, assuming the existence of the bases or the SVD.
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the_kid
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Homework Statement


A is arbitrary linear map of the complex vector space s.t. A: V-->V.

1) Show that there exist unitary matrices s.t A=V*DU where D is diagonal and its entires are non-negative.
2) Show that part 1 holds if and only if there exist orthonormal bases {u_i} and {v_i} s.t. Au_i=d_i v_i and A*v_i=d_i u_i where u_i is the i-th column of U* and v_i is the i-th column of V*.

Homework Equations


The Attempt at a Solution


I was able to successful prove part 1. However, I am really stuck on part 2. I know I need to show both directions since it is an iff proof. So, if I assume these bases exist, I need to be able to derive the SVD.
 
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  • #2
If I assume the SVD exists, I need to be able to derive the bases. I am not sure how to do either one of these. Any help would be greatly appreciated.
 

1. What is Singular Value Decomposition (SVD)?

Singular Value Decomposition (SVD) is a mathematical method that decomposes a matrix into three separate matrices, in order to simplify and analyze the original matrix. It is commonly used in data analysis, signal processing, and image recognition.

2. What is the purpose of using SVD?

SVD is used to identify patterns and relationships within a matrix, by breaking it down into smaller, more easily interpreted components. It also allows for dimensionality reduction, which can improve the efficiency and accuracy of certain algorithms.

3. What are the steps involved in SVD?

The first step in SVD is to decompose the original matrix into three matrices: U, Σ, and V*. U represents the left singular vectors, Σ represents a diagonal matrix of singular values, and V* represents the right singular vectors. The second step is to analyze and interpret the decomposed matrices, in order to gain insights into the original matrix.

4. How is SVD different from other matrix decomposition methods?

SVD is unique because it works on both square and non-square matrices, and can handle both real and complex numbers. It also allows for dimensionality reduction, unlike other decomposition methods such as LU decomposition or QR decomposition.

5. Can SVD be used for any type of matrix?

Yes, SVD can be used for any type of matrix, including sparse or dense matrices. However, it may not always be the most efficient or accurate method for certain types of matrices, and other decomposition methods may be more suitable.

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