Eigenvalues and Norms: Showing Existence of a Nonsingular Matrix

[xeno_gear]

Homework Statement

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

Homework Equations

[tex] \| \cdot \|_2 [/tex] is the induced 2-norm.

The Attempt at a Solution

Not much.. I know that [tex] \rho[/tex] is the spectral radius, and as such is equal to the infimum of all the (induced) norms of [tex] A[/tex]. Also, I know that [tex] A[/tex] and [tex] X^{-1}AX[/tex] 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..

 

[hunt_mat]

Try and diagonalise you matrix A, and then use [tex]P[/tex] as the matrix of eigenvectors, and look at [tex]X=P\cdot U[/tex] then you can look at diagonal matrices, then i think it should be just a matter of differentiation.

I could be wrong though.

 

[xeno_gear]

[tex] A [/tex] is arbitrary and isn't necessarily diagonalizable. There's an SVD and a Schur factorization though.

 

[dirk_mec1]

Use the Jordan Normal Form Theorem:

http://en.wikipedia.org/wiki/Spectral_radius

 

[dirk_mec1]

Forget my suggestion: it doesn't work.