Finding A Matrix, given eigenvalues, and eigenvectors

[Laura1321412]

Find a matrix that has eigenvalues 0,18,-18 with corresponding eigenvectors (0,1,-1), (1,-1,1), (0,1,1).

... I know the diagonlize rule, and the the rule to find a a power of A

A= PDP^-1
D=P^-1AP


... but i am lost as to how to contine... help please?

 

[arkajad]

Call you eigenvectors u1,u2,u3. Call your eigenvalues l1,l2,l3. Call you matrix A. I guess A is 3x3, so it has 9 coefficients.

You have 3 vector equations

Au1=l1u1
Au2=l2u2
Au3=l3u3

Consider the matrix coefficients a11,a12,a13, etc as unknowns. You have 3x3=9 linear equations for nine unknowns. Not too bad. There is a hope.

Of course there are other, more elegant methods. But this one is pretty straightforward.

 

[HallsofIvy]

It's not clear to me what you are asking. You say you have found the eigenvalues to be 0, 18, -18 with corresponding eigenvectors (0,1,-1), (1,-1,1), (0,1,1).

So your matrix P is given by
$$P= \begin{bmatrix} 0 & 1 & 0 \\ 1 & -1 & 1 \\ -1 & 1 & 1\end{bmatrix}$$
and D is
$$D= \begin{bmatrix}0 & 0 & 0 \\ 0 & 18 & 0 \\ 0 & 0 & -18\end{bmatrix}$$

Yes, it is true that $A= PDP^{-1}$, $D= P^{-1}AP$, and that $A^n= PD^nP^{-1}$.

Can you find $P^{-1}$?

It is
$$P^{-1}= \begin{bmatrix}1 & \frac{1}{2} & -\frac{1}{2} \\ 1 & 0 & 0 \\0 & \frac{1}{2} & \frac{1}{2}\end{bmatrix}$$.