# Find Matrix A from eigenvalues and eigenvectors?

## Homework Statement

Matrix A has eigenvalues $$\lambda$$1= 2 with corresponding eigenvector v1= (1, 3) and $$\lambda$$2= 1 with corresponding eigenvector v2= (2, 7), find A.

## Homework Equations

Definition of eigenvector: Avn=$$\lambda$$nvn

## The Attempt at a Solution

I tried this by making matrix A equal to:[ a, b, c, d ] (2x2 matrix) and then setting
v1(A - I*$$\lambda$$1) = v2(A - I*$$\lambda$$2)
(where I is the 2x2 identity matrix) and solving for a,b,c,d but it was wrong! Can anyone help?

$$\left[ \begin{array}{cccc} a_{1,1} & a_{1,2} \\ a_{2,1} & a_{2,2} \end{array} \right] \cdot \left[ \begin{array}{cccc} 1 & 2 \\ 3 & 7 \end{array} \right] = \left[ \begin{array}{cccc} 2 & 2 \\ 6 & 7 \end{array} \right]$$

I would set it up like so and then solve. I think your method is fine too, but more prone to algebraic mistake as you have demonstrated.

lanedance
Homework Helper
of how about normalising the eignvectors, then
A = V^T.D.V

where V is the matrix of normalised eigenvectors, D is the diagonal matrix of eignevalues