Find Matrix P and the Diagonal Matrix D

[chwala]

Homework Statement

see attached - refer to part b only (part a was easy)

Relevant Equations

matrices

For part (b) i was able to use equations to determine the eigenvectors;

For example for $λ =6$

$12x +5y -11z=0$
$8x-4z=0$
$32x+10y-26z=0$ to give me the eigen vector,


$\begin{pmatrix} 1 \\ 2 \\ 2 \end{pmatrix}$ and so on.

My question is to get matrix P does the arrangement of the eigenvector matrices matter?

In my arrangement for eigenvectors for $λ=6,-4,2$

i have,

$P=\begin{pmatrix} 1 & 1& 1 \\ 2 & 0 & -1 \\ 2 & 2 & 1 \end{pmatrix}$

and my Diagonal matrix is

$D=\begin{pmatrix} 6^5 & 0 & 0 \\ 0 & -4^5 & 0 \\ 0 & 0 & 2^5 \end{pmatrix}= \begin{pmatrix} 7776 & 0 & 0 \\ 0 & -1024 & 0 \\ 0 & 0 & 32 \end{pmatrix}$

 

[Mark44]

My question is to get matrix P does the arrangement of the eigenvector matrices matter?

No, but where you place the eigenvectors (i.e., in which columns) will determine P, which will determine $P^{-1}$ which will then determine where the eigenvalues appear on the diagonal matrix D.

 

[chwala]

No, but where you place the eigenvectors (i.e., in which columns) will determine P, which will determine $P^{-1}$ which will then determine where the eigenvalues appear on the diagonal matrix D.

Ok bass (boss). Gday @Mark44