# Find $3\times 3$ matrix when eigenvectors and eigenvalues are known.

## Problem

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

## Attempt to solve

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

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

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

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

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

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

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.

• Have you heard of diagonalization? – MisterRiemann Nov 25 '18 at 21:27

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

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

• $$Ax_1=\lambda_1x_1$$
• $$Ax_2=\lambda_2x_2$$
• $$Ax_3=\lambda_3x_3$$