1. ## Matrices and eigenvalues

Assume that A(t) is a 2x2 matrix with real, distinct eigenvalues, lam1 < lam2.

Let B denote the set of 2x2 matrices with eigenvalues of negative real part such that xBx > 0 for some x in R2. Prove that if ||x(t)|| --> infinity as t --> infinity, then A(t) in B for some t > 0.

I'm not sure where to start with this one. Any help would be greatly appreciated!

2. Originally Posted by hgrin123
Assume that A(t) is a 2x2 matrix with real, distinct eigenvalues, lam1 < lam2.

Let B denote the set of 2x2 matrices with eigenvalues of negative real part such that xBx > 0 for some x in R2. Prove that if ||x(t)|| --> infinity as t --> infinity, then A(t) in B for some t > 0.

I'm not sure where to start with this one. Any help would be greatly appreciated!

In what normed linear space are you working and what norm is defined there? What is x(t)? What is A(t)? Is this a matrix that depends HOW on a parameter t? And how $||x(t)|| \xrightarrow [t\to \infty] {} \infty$ implies anything at all for A(t)?

Tonio