Find Matrix A from eigenvalues and eigenvectors?

[hotrokr69]

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: Av_n=$$\lambda$$_nv_n

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?

 

[PhDorBust]

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

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]

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