Matrix Decomposition Explained: Simple Illustration

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Discussion Overview

The discussion revolves around the concept of matrix decomposition, with participants seeking a simple illustration and clarification of its types and applications. The scope includes theoretical aspects and potential applications in fields like elasticity.

Discussion Character

  • Exploratory
  • Conceptual clarification
  • Debate/contested

Main Points Raised

  • Some participants request a simple illustration of matrix decomposition, indicating a need for clarity on the topic.
  • Others mention various types of matrix decomposition, such as LU-decomposition and QR-decomposition, suggesting a range of methods available.
  • One participant emphasizes the importance of specifying whether the interest lies in the reasons for performing decomposition or the methods of performing it.
  • Singular Value Decomposition (SVD) is noted as a popular method, with a general idea that decomposition simplifies complex matrices into components.
  • A participant relates matrix decomposition to elasticity, specifically mentioning the deformation tensor and its relation to stretch and rotation tensors, but expresses uncertainty about the specific type of decomposition used.
  • Polar decomposition is introduced as a specific type of decomposition that separates a matrix into an orthogonal rotation and a symmetric positive definite stretching.
  • Another participant suggests that shearing maps may also be relevant to the discussion.

Areas of Agreement / Disagreement

Participants express varying levels of understanding and interest in different types of matrix decomposition, with no consensus reached on a specific method or application. Some participants seek clarification while others provide links and references, indicating a mix of knowledge levels and perspectives.

Contextual Notes

There are limitations in the specificity of the questions posed, as well as the definitions and types of decomposition discussed. The discussion does not resolve which type of decomposition is most applicable to the context of elasticity.

mohammed El-Kady
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Can anyone illustrate for me matrix decomposition in a simple way?
 
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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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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.
 
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
 
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
 
You may also be interested in shearing maps.
 

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