- #1

- 36

- 0

## Homework Statement

Let A be a 5x5 matrix over R

3 eigenvectors of A are:

[itex]u_1=(1,0,0,1,1)[/itex]

[itex]u_2=(1,1,0,0,1)[/itex]

[itex]u_3=(-1,0,1,0,0)[/itex]

also:

[itex]\rho (2I-A)>\rho (3I-A)[/itex]

and: [itex]A(1,2,2,1,3)^t=(0,4,6,2,6)^t[/itex]

prove that A is diagonalizable and find a diagonal matrix similiar to it.

## Homework Equations

## The Attempt at a Solution

What I can make of this is:

[itex](1,2,2,1,3)^t=u_1 + 2u_2 + 2u_3[/itex]

(then maybe I can say that P has the eigenvectors as columns and

[itex]AP(1,2,2,0,0)^t=(0,4,6,2,6)^t[/itex]

but then what?)

and because

[itex]\rho (2I-A)>\rho (3I-A)[/itex]

[itex]\rho (2I-A)\leq 5[/itex]

it means that

[itex]5>\rho (3I-A)[/itex]

and so

[itex]det(3I-A)=0[/itex]

and 3 is an eigenvalue of A.

I would like some hints/suggestions on what to do.

Thanks!