Singular Value Decomposition of an nxn matrix?

In summary, Singular Value Decomposition (SVD) is a matrix factorization technique used in various applications such as data compression, image processing, and machine learning. Its purpose is to simplify complex matrices and identify patterns within the data. SVD differs from other factorization methods in its ability to be applied to any type of matrix and produce the most compact representation. It has a wide range of applications in fields such as image and signal processing, natural language processing, and bioinformatics. SVD is calculated by finding the eigenvalues and eigenvectors of the original matrix and using them to construct the left and right singular matrices, as well as the diagonal matrix of singular values, through various algorithms.
  • #1
s_j_sawyer
21
0
I was just wondering if it was possible to find the singular value decomposition of an nxn matrix such as

1 1
-1 1

I tried this but then when finding the eigenvectors of A^T*A I found there were none (non-trivial anyhow).

So, is this not possible?

EDIT:

How embarrassing I made an error in my calculations. Sorry, it's all good now.
 
Last edited:
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  • #2
What do you think A^T*A is? It looks to me like it has lots of eigenvectors.
 

1. What is Singular Value Decomposition (SVD)?

SVD is a matrix factorization technique that breaks down a matrix into three components - a left singular matrix, a diagonal matrix of singular values, and a right singular matrix. This decomposition is useful in many applications, including data compression, image processing, and machine learning.

2. What is the purpose of performing SVD on a matrix?

The purpose of SVD is to simplify complex matrices and make them easier to work with. It can also help identify patterns and relationships within the data, which can be useful in data analysis and modeling.

3. How does SVD differ from other matrix factorization methods?

SVD is unique in that it can be applied to any type of matrix, regardless of its size or shape. It also produces the most compact representation of the original matrix, making it useful for data compression. Other factorization methods may only work on specific types of matrices or may not produce as concise results.

4. What are the applications of SVD?

SVD has a wide range of applications in various fields, including image and signal processing, natural language processing, recommendation systems, and bioinformatics. It is also commonly used in data analysis and machine learning algorithms.

5. How is SVD calculated?

The calculation of SVD involves finding the eigenvalues and eigenvectors of the original matrix. These are then used to construct the left and right singular matrices, and the diagonal matrix of singular values. There are various algorithms for calculating SVD, including the power method and the Golub-Reinsch algorithm.

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