Eigenvalues and Norms: Showing Existence of a Nonsingular Matrix

  • Thread starter Thread starter xeno_gear
  • Start date Start date
  • Tags Tags
    Eigenvalues
Click For Summary

Homework Help Overview

The problem involves a matrix \( A \) in \( \mathbb{C}^{n \times n} \) and requires showing the existence of a nonsingular matrix \( X \) such that the induced 2-norm of the transformed matrix \( X^{-1}AX \) is bounded by the spectral radius \( \rho \) plus an arbitrary small positive value \( \varepsilon \).

Discussion Character

  • Exploratory, Assumption checking

Approaches and Questions Raised

  • Participants discuss the properties of the spectral radius and its relationship to induced norms. There is mention of diagonalization and the use of matrices of eigenvectors, but concerns are raised about the diagonalizability of \( A \). Alternative approaches such as Singular Value Decomposition (SVD) and Schur factorization are also suggested.

Discussion Status

The discussion is ongoing, with various approaches being explored. Some participants have offered suggestions regarding diagonalization and matrix transformations, while others have questioned the applicability of these methods given the arbitrary nature of matrix \( A \). There is no clear consensus yet, but multiple lines of reasoning are being examined.

Contextual Notes

Participants note that \( A \) is not necessarily diagonalizable, which may affect the methods considered. The discussion includes references to the Jordan Normal Form and the spectral radius, indicating a focus on the properties of matrices in relation to eigenvalues.

xeno_gear
Messages
40
Reaction score
0

Homework Statement


Let A \in \mathbb{C}^{n \times n} and set \rho = \max_{1 \le i \le n}|\lambda_i|, where \lambda_i \, (i = 1, 2, \dots, n) are the eigenvalues of A. Show that for any \varepsilon > 0 there exists a nonsingular X \in \mathbb{C}^{n \times n} such that \|X^{-1}AX\|_2 \le \rho + \varepsilon.


Homework Equations


\| \cdot \|_2 is the induced 2-norm.


The Attempt at a Solution


Not much.. I know that \rho is the spectral radius, and as such is equal to the infimum of all the (induced) norms of A. Also, I know that A and X^{-1}AX have the same eigenvalue properties (eigenvalues, spectral radius, algebraic and geometric multiplicities) since they're similar matrices. I can't quite figure out how to use these though. Any thoughts? Thanks..
 
Physics news on Phys.org
Try and diagonalise you matrix A, and then use P as the matrix of eigenvectors, and look at X=P\cdot U then you can look at diagonal matrices, then i think it should be just a matter of differentiation.

I could be wrong though.
 
A is arbitrary and isn't necessarily diagonalizable. There's an SVD and a Schur factorization though.
 
Forget my suggestion: it doesn't work.
 

Similar threads

Does each norm on vector space become discontinuous when restricted to S^1?
  • · Replies 1 ·
Replies
1
Views
1K
Can Matrix Norms be Used to Bound the Eigenvalues of a Matrix?
  • · Replies 1 ·
Replies
1
Views
2K
Proving Unitary Matrix Norm: $$||UA||_2 = ||AU||_2$$
  • · Replies 12 ·
Replies
12
Views
4K
Matrix with a bounded mapping as an entry is bounded
  • · Replies 2 ·
Replies
2
Views
1K
Show that given n distinct eigenvalues, eigenvectors are independent
  • · Replies 1 ·
Replies
1
Views
2K
Show operator is compact/symmetric
  • · Replies 5 ·
Replies
5
Views
2K
Inverse of diagonal dominant matrices
  • · Replies 2 ·
Replies
2
Views
2K
Show that a space is a Banach space
  • · Replies 12 ·
Replies
12
Views
2K
Showing Uniform Convergence of Cauchy Sequence of Functions
  • · Replies 6 ·
Replies
6
Views
3K
Showing That the Eigenvalue of a Matrix is 0
  • · Replies 8 ·
Replies
8
Views
2K