Question

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

Answer 1

Let A =
\[\left[\begin{matrix}1 & 0 & 0 \\ 0 & 0 & 0 \\ 4 & 16 & -1\end{matrix}\right]\]

Answer 2

then, find \(\lambda\)

Answer 3

\[[A-\lambda]=\left[\begin{matrix}1-\lambda & 0 & 0 \\ 0 & -\lambda & 0 \\ 4 & 16 & -1-\lambda\end{matrix}\right]

\]
then, the value of \(\lambda\) are 0, 1, -1
then, P is matrix of eigenvectors
and D is \(\left[\begin{matrix}1 & 0 & 0 \\ 0 & 0 & 0 \\ 0 & 0 & -1\end{matrix}\right]\)

Answer 4

I get P = \[\left[\begin{matrix}1/2 & 0 & 0 \\ 0 & 1/16 & 0 \\ 1 & 1 & 1 \end{matrix}\right]\]
and it says it is wrong. Where is my mistake?

Answer 5

The eigenvectors are \(\{(0, 0, 1), (1, 0, 2), (0, 1, 16)\}\)

Answer 6

:) you need show your work to see where the mistake is.

Answer 7

Remember that to obtain the eigenvectors corresponding to eigenvalue \(\lambda\) you are solving for the nullspace of \(A - \lambda I\).

Answer 8

The nullspace consists of the solutions to the following equation.
\((A - \lambda I)x = 0\)