Matrix Decomposition Explained: Simple Illustration

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SUMMARY

Matrix decomposition is a mathematical technique used to simplify complex matrices into more manageable components. Key types include LU-decomposition, QR-decomposition, and Singular Value Decomposition (SVD). The polar decomposition specifically breaks a matrix into an orthogonal rotation matrix (U) and a symmetric positive definite stretching matrix (P). Understanding these decompositions is essential for applications in elasticity, particularly in analyzing deformation tensors.

PREREQUISITES
  • Matrix algebra fundamentals
  • Understanding of LU-decomposition
  • Familiarity with QR-decomposition
  • Knowledge of Singular Value Decomposition (SVD)
NEXT STEPS
  • Study the applications of LU-decomposition in solving linear equations
  • Explore QR-decomposition for least squares problems
  • Learn about Singular Value Decomposition (SVD) and its role in data compression
  • Investigate the polar decomposition and its implications in elasticity
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Mathematicians, engineers, and data scientists who require a deeper understanding of matrix operations and their applications in fields such as elasticity and data analysis.

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
 
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