Jordan Decomposition to Schur Decomposition

In summary, you can obtain a Schur Decomposition from a Jordan Decomposition by using SVDs or QR decompositions of X or J. This can be done because Q is orthogonal and Q*=Q-1. However, if you are still having trouble, it is recommended to seek further suggestions for a solution.
  • #1
azdang
84
0

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.
 
Physics news on Phys.org
  • #2
No ideas? Sorry to be impatient. I just really am not getting anywhere.
 

1. What is the Jordan decomposition to Schur decomposition?

The Jordan decomposition to Schur decomposition is a mathematical technique used to decompose a square matrix into two matrices - a Jordan matrix and a Schur matrix. This technique is often used in linear algebra to simplify calculations and solve problems related to matrices.

2. How is the Jordan decomposition to Schur decomposition performed?

The Jordan decomposition to Schur decomposition is performed by first finding the eigenvalues and corresponding eigenvectors of the matrix. The eigenvalues are then used to construct the Jordan matrix, which contains the eigenvalues along the diagonal and ones in the upper diagonal. The Schur matrix is then obtained by subtracting the Jordan matrix from the original matrix.

3. What is the purpose of using the Jordan decomposition to Schur decomposition?

The purpose of using the Jordan decomposition to Schur decomposition is to simplify a matrix into two matrices that are easier to work with. The Jordan matrix has a simpler structure, making it easier to calculate its powers and inverse. The Schur matrix is upper triangular, which makes it easier to solve systems of equations involving the matrix.

4. Can the Jordan decomposition to Schur decomposition be applied to any matrix?

No, the Jordan decomposition to Schur decomposition can only be applied to square matrices. Additionally, not all square matrices have a Jordan decomposition. A matrix must have a complete set of eigenvectors in order for the Jordan decomposition to exist.

5. What are some applications of the Jordan decomposition to Schur decomposition?

The Jordan decomposition to Schur decomposition has various applications in mathematics and engineering. It is used to solve systems of differential equations, to diagonalize matrices, and to find the exponential of a matrix. It is also used in control theory, signal processing, and data compression.

Similar threads

  • Linear and Abstract Algebra
Replies
1
Views
913
  • Calculus and Beyond Homework Help
Replies
1
Views
3K
  • Calculus and Beyond Homework Help
Replies
1
Views
1K
Replies
1
Views
2K
  • Calculus and Beyond Homework Help
Replies
6
Views
3K
  • Linear and Abstract Algebra
Replies
1
Views
1K
  • Calculus and Beyond Homework Help
Replies
2
Views
3K
Replies
2
Views
5K
  • Calculus and Beyond Homework Help
Replies
7
Views
2K
  • Programming and Computer Science
Replies
5
Views
3K
Back
Top