Question

A = [5,8/3,-2/3; 2,2/3,4/3; -4, -4/3, -8/3]

(Matrix form 3x3)

How do I write down a matrix that diagonalises A? Please show all working.

Answer 1

find the eigenvectors

Answer 2

yep i have got them whats next?

Answer 3

form a matrix P
then \[P^{-1}AP\] is a diagonal matrix

Answer 4

so basically form a 3x3 out of the eigenvectors

Answer 5

yes

Answer 6

Zarkon would you mind giving this a go nd posting your solutions?

Answer 7

i know the eigenvalues are:

lambda = 0
lambda = -3
lambda = 6

Answer 8

correct

Answer 9

I get \[P=\left[\begin{matrix}-2 & 1/4 & -2 \\ -1/2 & -1/2 & 4\\ 1 & 1& 1\end{matrix}\right]\]

then \[P^{-1}AP=\left[\begin{matrix}6 & 0 & 0 \\ 0 & -3 & 0\\ 0 & 0& 0\end{matrix}\right]\]

Answer 10

yes that is correct although what are the workings for it.

Answer 11

Compute the null space for the matrices

\[A-\lambda I\]

\[\text{Where }\lambda=6,-3,0\]

Answer 12

ok thanks i will keep going you are too good.

Answer 13

do I need to have an eigenvector for when lambda is 0?

Answer 14

yes