Eigenvalues and Norms: Showing Existence of a Nonsingular Matrix

[xeno_gear]

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..

 

[hunt_mat]

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.

 

[xeno_gear]

$$A$$ 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.