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