Polar and Jordan Decomp. in Intro to Linear Algebra?

In summary, the conversation discusses the topics of polar decomposition and Jordan forms in the context of an undergraduate level Intro to LA course. The speaker acknowledges that they have a basic understanding of these concepts and can apply formulas to solve simple exercises, but wonders if the course should go deeper into these topics or if they should look for outside resources. The speaker also mentions their free notes that explain the Jordan form in three different courses and provides a link to a resource on polar decomposition.
  • #1
kostoglotov
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My Intro to LA course has visited the ideas of polar decomposition and Jordan forms, but not gone into them in depth. I wouldn't say I understood them, but I'm aware of them, and could possibly solve some basic exercises involving them if all I had to do was apply formulas.

My question is: should an undergraduate level Intro to LA course (in the context of supporting an overall Electrical Engineering degree) go deeper into Polar and Jordan Decomposition? Should I look for some outside resources to fill out my understanding in these areas, or leave it til it becomes clearer whether I'll need it?
 
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  • #2
If you want more notes my free ones explain the jordan form in three different courses, math 845, 8000, and 4050. (I never needed nor learned the polar form, but when i do i will maybe write it up too.)

http://alpha.math.uga.edu/~roy/
 
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Related to Polar and Jordan Decomp. in Intro to Linear Algebra?

1. What is the Polar Decomposition in Intro to Linear Algebra?

The Polar Decomposition in Intro to Linear Algebra is a theorem that states that any square matrix can be decomposed into a product of two matrices, one of which is orthogonal and the other is positive definite. This decomposition is useful for understanding the geometric properties of matrices and their transformations.

2. What is the Jordan Decomposition in Intro to Linear Algebra?

The Jordan Decomposition in Intro to Linear Algebra is a theorem that states that any square matrix can be decomposed into a sum of a diagonal matrix and a nilpotent matrix. This decomposition is useful for understanding the algebraic properties of matrices and solving systems of linear equations.

3. How are the Polar and Jordan Decompositions related?

The Polar and Jordan Decompositions are related by the spectral theorem, which states that a symmetric matrix can be decomposed into a product of an orthogonal matrix and a diagonal matrix. This decomposition is a special case of both the Polar and Jordan Decompositions.

4. Can the Polar and Jordan Decompositions be applied to non-square matrices?

No, the Polar and Jordan Decompositions are only defined for square matrices. However, there are similar decompositions for non-square matrices, such as the singular value decomposition for rectangular matrices.

5. How are the Polar and Jordan Decompositions used in practical applications?

The Polar and Jordan Decompositions have many practical applications, such as in signal processing, image compression, and solving differential equations. They are also used in various areas of engineering and physics, such as in quantum mechanics and control theory.

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