Find Matrix A from eigenvalues and eigenvectors?

  • Thread starter hotrokr69
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  • #1
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Homework Statement



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


Homework Equations



Definition of eigenvector: Avn=[tex]\lambda[/tex]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*[tex]\lambda[/tex]1) = v2(A - I*[tex]\lambda[/tex]2)
(where I is the 2x2 identity matrix) and solving for a,b,c,d but it was wrong! Can anyone help?
 

Answers and Replies

  • #2
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[tex]
\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]
[/tex]

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.
 
  • #3
lanedance
Homework Helper
3,304
2
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
 

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