# Homework Help: Linear algebra- diagonalizability

1. Dec 11, 2011

### foxofdesert

1. The problem statement, all variables and given/known data
Compute A17.

1 0 4
A=0 1 2
0 0 4

2. Relevant equations

A=PDP-1 (D is a diagonal matrix)

3. The attempt at a solution

I got eigenvalues 1 and 4, and corresponding eigenspaces u1=[1,0,0]T , u2=[0,1,1]T and u3=[4/3, 2/3 ,1]T.

So, I computed P= (1/3) 3 0 4
0 3 2
0 0 3

also, P-1= (1/3) 3 0 -4
0 3 -2
0 0 3.

D= 1/3 1 0 0
0 1 0
0 0 4.

and then A17= PD17P-1

= 1/9 3 0 4$\cdot$417 3 0 -4
0 3 2$\cdot$417 0 3 -2
0 0 3$\cdot$417 0 0 3

= 1 0 4/3(417-1)
0 1 2/3(417-1)
0 0 417

sorry for the matrix form here. I couldn't find any commands to do it correctly.

Am I doing right here?

2. Dec 11, 2011

### foxofdesert

oh... I cannot put spaces correctly :( everything here is 3x3 matrices.

3. Dec 11, 2011

### Deveno

to display matrices correctly use latex. for example,

begin{bmatrix}a&b&c\\d&e&f\\g&h&i\end{bmatrix} preceded by a slash,

which gives the display:

$$\begin{bmatrix}a&b&c\\d&e&f\\g&h&i\end{bmatrix}$$

now if A = PDP-1, where D is diagonal,

then A17 = PD17P-1

and D17 is easy to compute, we just take the 17th power of the diagonal elements.

it looks like you have calculated P correctly, but you wrote down the wrong eigenvector for u2, it should be [0,1,0]T. P-1, also looks ok to me.

however, i get that D should just be:

$$\begin{bmatrix}1&0&0\\0&1&0\\0&0&4\end{bmatrix}$$

which is going to throw off your values for A17