Jordan Decomposition to Schur Decomposition

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SUMMARY

The discussion focuses on deriving a Schur Decomposition from a Jordan Decomposition for a square matrix A, represented as A = XJX-1, where X is non-singular and J is upper bidiagonal. The Schur Decomposition is defined as A = QTQ*, with Q being unitary or orthogonal and T being upper triangular. Participants express confusion regarding the relationship between the two decompositions and the role of orthogonality in the process.

PREREQUISITES
  • Understanding of Jordan Decomposition
  • Familiarity with Schur Decomposition
  • Knowledge of matrix properties, specifically unitary and orthogonal matrices
  • Basic concepts of linear algebra, including eigenvalues and eigenvectors
NEXT STEPS
  • Study the properties of Jordan Decomposition in detail
  • Learn about Schur Decomposition and its applications in numerical analysis
  • Explore the relationship between unitary matrices and orthogonal matrices
  • Investigate Singular Value Decomposition (SVD) and its connection to other matrix decompositions
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Students and professionals in mathematics, particularly those studying linear algebra, matrix theory, and numerical methods. This discussion is beneficial for anyone looking to deepen their understanding of matrix decompositions.

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Homework Statement


Let A be a complex or real square matrix. Suppose we have a Jordan decomposition A = XJX-1, where X is non-singular and J is upper bidiagonal. Show how you can obtain a Schur Decomposition from a Jordan Decomposition.


Homework Equations


Schur Decomposition: A = QTQ*, where Q is unitary/orthogonal and T is upper triangular with the eigenvalues of A on the diagonal.


The Attempt at a Solution


I'm really not sure at all what to do. Because Q is orthogonal, Q*=Q-1. I'm not sure if that plays in somehow.

I've been trying to use SVDs or QR decompositions of X or J to get there, but I've had no luck. Does anyone have any suggestions? Thank you so much.
 
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No ideas? Sorry to be impatient. I just really am not getting anywhere.
 

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