Matrix Decomposition Explained: Simple Illustration

In summary, matrix decomposition is the process of breaking down a matrix into simpler components. There are various types of decomposition, such as LU-decomposition, QR-decomposition, and SVD. The idea behind decomposition is to simplify the matrix by breaking it into smaller parts. In the context of elasticity, matrix decomposition can also be used to prove the relationship between the deformation tensor and the stretch and rotation tensors. One example of decomposition in elasticity is the polar decomposition, which breaks a matrix into an orthogonal rotation and a symmetric positive definite stretching. Another related concept is shearing maps.
  • #1
mohammed El-Kady
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Can anyone illustrate for me matrix decomposition in a simple way?
 
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  • #3
mohammed El-Kady said:
Can anyone illustrate for me matrix decomposition in a simple way?

There are several types: LU-decomposition, QR- decomposition, and probably others.

You need to be more specific, and your question needs more focus. Are you interested in (1) WHY perform decompositition; or (2) HOW to perform decompostition?
 
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  • #5
thank you for your responses, while i study elasticity it have been mentioned that deformation tensor is stretch and rotation tensor and the proof by using matrix decomposition, I've no idea about the type of decomp.
 
  • #6
mohammed El-Kady said:
thank you for your responses, while i study elasticity it have been mentioned that deformation tensor is stretch and rotation tensor and the proof by using matrix decomposition, I've no idea about the type of decomp.
It's the decomposition into a orthogonal rotation ##U## and a symmetric positive definite stretching ##P##, see the polar decomposition https://en.wikipedia.org/wiki/Polar_decomposition
 
  • #7
fresh_42 said:
It's the decomposition into a orthogonal rotation ##U## and a symmetric positive definite stretching ##P##, see the polar decomposition https://en.wikipedia.org/wiki/Polar_decomposition
thank you too much, its helpful and valuable and easy
 
  • #8
You may also be interested in shearing maps.
 

1. What is matrix decomposition?

Matrix decomposition, also known as matrix factorization, is a mathematical process that breaks down a complex matrix into smaller, simpler matrices. This allows for easier manipulation and analysis of the data within the matrix.

2. What are the different types of matrix decomposition?

There are several types of matrix decomposition, including LU decomposition, QR decomposition, and singular value decomposition (SVD). Each type has its own unique approach and application.

3. How is matrix decomposition used in data analysis?

Matrix decomposition is commonly used in data analysis to reduce the dimensionality of a dataset, identify patterns and relationships between variables, and make predictions based on the decomposed matrix.

4. Can you provide a simple illustration of matrix decomposition?

Imagine you have a matrix that represents the grades of students in a class. By decomposing the matrix, you can identify which students are performing well in which subjects, and group them accordingly. This can help with identifying areas of improvement and overall performance of the class.

5. What are the advantages of using matrix decomposition?

The main advantage of matrix decomposition is that it simplifies complex data and allows for easier analysis and interpretation. It also helps to reduce the number of variables and can improve the accuracy of predictions and models. Additionally, different types of matrix decomposition have different advantages, making it a versatile tool for various applications.

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