Linear algebra- diagonalizability

[foxofdesert]

Homework Statement

Compute A¹⁷.

1 0 4
A=0 1 2
0 0 4

Homework Equations

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

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\cdot4¹⁷ 3 0 -4
0 3 2\cdot4¹⁷ 0 3 -2
0 0 3\cdot4¹⁷ 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?

 

[foxofdesert]

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

 

[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