How to find invertible matrix and diagonal matrix

[shellizle]

Homework Statement

Find an invertible matrix P and a diagonal matrix D such that D=P^(-1)AP.

A=
(−1 2 −2)
(−2 3 −2)
( 2 −2 3)

Homework Equations

The Attempt at a Solution

the eigenvalues are 1, 1, and 3
the eigenvector I've found so far is for the eigenvalue 3, which is t(1, -1, 1)
when i was attempting to find the eigenvector of 1,

i did (1*I-A)(x), and then reduced the matrix to

(1 1 -1)
(0 0 0)
(0 0 0)
and in polynomial form it would be
x+y-z=0

how am i suppose to put this in homogeneous form? x=(a, b, c)t

thanks in advance!

 

[Mark44]

Homework Statement

Find an invertible matrix P and a diagonal matrix D such that D=P^(-1)AP.

A=
(−1 2 −2)
(−2 3 −2)
( 2 −2 3)

Homework Equations

The Attempt at a Solution

the eigenvalues are 1, 1, and 3
the eigenvector I've found so far is for the eigenvalue 3, which is t(1, -1, 1)
when i was attempting to find the eigenvector of 1,

i did (1*I-A)(x), and then reduced the matrix to

(1 1 -1)
(0 0 0)
(0 0 0)
and in polynomial form it would be
x+y-z=0

how am i suppose to put this in homogeneous form? x=(a, b, c)t

thanks in advance!

Your equation x + y + z = 0 will give you two solutions that correspond to the two eigenvectors you need for this eigenvalue.

x = - y - z
y = y
z = ...z

So (x, y, z) = y(-1, 1, 0) + z(-1, 0, 1)

y and z are parameters. For one eigenvector, choose y = 1, z = 0. For the other, choose y = 0, z = 1.

 

[shellizle]

yeah i tried that,
our assignment is online based and it tells you immediately if its right or wrong..
however I've tried everything but nothing works!
does the order for the eigenvalues matter? (x1, x2,and x3)
does the engevectors have to be alligned in a specific way for A=PDP^-1 to be valid?

 

[Mark44]

The order of the columns in your matrix P (and hence P^-1) determine where the eigenvalues appear where in your diagonal matrix. Is that what you're asking?