Singular Value Decomposition question

In summary: V1(hermitic); U2*Σ2*V2(hermitic)] = [U1*Σ1*V1(hermitic); U2*Σ2*V2(hermitic)]This shows that Σ = [Σ1; Σ2] and therefore, we can express Σ as Σ = [Σ1; Σ2].In summary, we can express the matrix Σ in the SVD formula as a concatenation of the individual Σ matrices of the blocks A1 and A2. I hope this provides some clarity on the topic. Good luck with your studies!
  • #1
juansg
1
0
Hi everybody and congrats for this forums!

I'm studying first course of Industrial Engineering, and last week we were asked in Algebra about the following question:

SVD: A=U*Σ*V(hermitic). If A is a matrix which we divide by an horizontal imagine line, so that we have two blocks: one in the upper section (A1), and the other (A2) below A1.
The question is, how to express Σ=Σ(Σ1,Σ2)

Σ1 is the Σ of A1, and Σ2 is the Σ of A2

I've been eating my brain this entire week to solve it, but I'm not able, so any answers, tips or anything would be really apreciated

thanks in advance

Juan
 
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  • #2


Hello Juan,

First of all, congratulations on studying Industrial Engineering! It's a fascinating field with many applications in various industries.

To answer your question, let's start by breaking down the SVD formula. As you mentioned, A=U*Σ*V(hermitic) can be divided into two blocks: A1 and A2. This means that A can be expressed as:

A = [A1; A2]

Where [A1; A2] represents the concatenation of A1 and A2. Now, let's look at the SVD formula again, but this time, we will divide it into two blocks as well:

U*Σ*V(hermitic) = [U1; U2] * [Σ1; Σ2] * [V1; V2](hermitic)

Where [U1; U2] represents the concatenation of U1 and U2, and the same goes for [V1; V2](hermitic). Now, let's simplify this further:

A = [A1; A2] = [U1; U2] * [Σ1; Σ2] * [V1; V2](hermitic)

Since A1 and A2 are both blocks of A, they can be expressed as A1 = U1*Σ1*V1(hermitic) and A2 = U2*Σ2*V2(hermitic). Substituting these values into the above equation, we get:

[U1*Σ1*V1(hermitic); U2*Σ2*V2(hermitic)] = [U1; U2] * [Σ1; Σ2] * [V1; V2](hermitic)

Now, since U and V are unitary matrices, their inverses are equal to their conjugate transpose, i.e. U(hermitic) = U^-1 and V(hermitic) = V^-1. Using this property, we can rewrite the above equation as:

[U1*Σ1*V1(hermitic); U2*Σ2*V2(hermitic)] = [U1; U2] * [Σ1; Σ2] * [V1; V2](hermitic)

[U1
 

1. What is Singular Value Decomposition (SVD)?

Singular Value Decomposition (SVD) is a mathematical technique used to decompose a matrix into three matrices: U, Σ, and V. It is commonly used in data analysis and machine learning to reduce the dimensionality of a dataset and identify patterns or relationships between variables.

2. How does SVD differ from other matrix decomposition methods?

SVD is unique because it can be applied to any type of matrix, including rectangular and sparse matrices. Other methods, such as LU decomposition, can only be applied to square matrices. SVD also provides the most accurate decomposition of a matrix, making it useful for data analysis and machine learning tasks.

3. What are the applications of SVD?

SVD has a wide range of applications, including image and signal processing, recommender systems, and natural language processing. It can also be used for data compression, noise reduction, and feature extraction.

4. How is SVD calculated?

The calculation of SVD involves finding the eigenvectors and eigenvalues of a matrix and then arranging them into the three matrices U, Σ, and V. This can be done using various algorithms, such as the power method or the QR algorithm.

5. What are the benefits of using SVD?

The main benefit of using SVD is its ability to reduce the dimensionality of a dataset while preserving the most important information. This can lead to improved data analysis and machine learning models. Additionally, SVD can handle missing values and noisy data, making it robust and versatile for various applications.

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