Ytukyg

Problem

Find $3times3$ matrix $textbf{A}$ that has eigenvalues of $lambda_1=-1,lambda_2=1,lambda_3=2$ and corresponding eignevectors $x_1=begin{bmatrix}0 \ 1 \ 2 end{bmatrix},x_2=begin{bmatrix}1 \ 1 \ 0end{bmatrix},x_3=begin{bmatrix}1 \2 \1end{bmatrix}$

Attempt to solve

I try to compute generic $3times 3$ matrix and solve it's eigenvalues. We can compute eigenvalues via characteristic polynomial which is defined as:

$$ P_A(lambda) := det(textbf{A}-lambdatextbf{I}) $$

$$ P_A(lambda)=det(begin{bmatrix} a & b & c \ d & e & f \ g & h & i end{bmatrix}-begin{bmatrix} lambda & 0 & 0 \ 0 & lambda & 0 \ 0 & 0 & lambda end{bmatrix}) $$
$$=begin{vmatrix} a-lambda & b & c \ d & e-lambda & f \ g & h & i-lambda end{vmatrix}$$

$$=(a-lambda)begin{vmatrix} e-lambda & f \ h & i-lambda end{vmatrix}-bbegin{vmatrix} d & f \ g & i-lambda end{vmatrix} + c begin{vmatrix} d & e-lambda \ g & h end{vmatrix} $$
$$ =(a-lambda)[(e-lambda)(i-lambda)-fh)]-b[d(i-lambda)-fg]+c[dh-g(e-lambda)] $$
$$ =(a-lambda)[ei+e(-lambda)-lambda i -lambda (-lambda)-fh]-b[di-dlambda-fg]+c[dh-ge-glambda] $$
$$= (a-lambda)[lambda^2-ilambda-elambda+ei]-b[-dlambda-di-fg]+c[-glambda-+dh-ge] $$

$$ = alambda^2-ialambda-ealambda+eai-lambda^3+ilambda^2+elambda^2-eilambda +bdlambda + bdi + bfg +cdh -cge-cglambda $$

$$ P_A(lambda) = -lambda^3+lambda^2(a+i+e)+lambda(-ia-ea-ei+bd-cg)+eai+bdi+bfg+cdh-cge $$

Eigenvalues can be found when $P_A(lambda)=0$. Only problem is I'am not quite sure if this is the best approach for this problem.

Any suggestion on how to proceed / change the approach all together.

$A=begin{pmatrix}0&1&1\1&1&2\2&0&1end{pmatrix}begin{pmatrix}-1&0&0\0&1&0\0&0&2end{pmatrix}begin{pmatrix}0&1&1\1&1&2\2&0&1end{pmatrix}^{-1}$.

HINT

Recall that for $A$ diagonalizable

$$A=PDP^{-1}$$

as an alternative refer to the definition of eigenvalues and, assuming as unknowns the 9 entries for $A$, solve

• $Ax_1=lambda_1x_1$


• $Ax_2=lambda_2x_2$


• $Ax_3=lambda_3x_3$