Find Matrix A from eigenvalues and eigenvectors?

T denotes the transpose of the matrix.In summary, to find matrix A given its eigenvalues and eigenvectors, one can use the formula A = V^T.D.V, where V is the matrix of normalised eigenvectors and D is the diagonal matrix of eigenvalues. Another method is to set up a system of equations and solve for the elements of A.
  • #1
hotrokr69
1
0

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?
 
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  • #2
[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
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
 

What is a matrix A?

A matrix is a rectangular array of numbers or symbols arranged in rows and columns. It is often used to represent linear transformations and solve systems of linear equations.

What are eigenvalues and eigenvectors?

Eigenvalues and eigenvectors are important concepts in linear algebra. Eigenvalues are scalar values that represent the scaling factor of eigenvectors, which are non-zero vectors that are only scaled by a linear transformation.

How do I find matrix A from given eigenvalues and eigenvectors?

To find matrix A from given eigenvalues and eigenvectors, you can use the formula A = PDP^-1, where P is a matrix whose columns are the eigenvectors of A, and D is a diagonal matrix with the corresponding eigenvalues on the diagonal. This is known as the diagonalization method.

Why is finding matrix A from eigenvalues and eigenvectors important?

Finding matrix A from eigenvalues and eigenvectors allows us to understand the behavior of linear transformations and solve systems of linear equations more efficiently. It also has applications in other areas such as computer graphics, data analysis, and quantum mechanics.

Are there other methods to find matrix A from eigenvalues and eigenvectors?

Yes, there are other methods such as the power method, which is an iterative algorithm that uses the dominant eigenvalue and eigenvector to approximate matrix A. There is also the Jordan canonical form, which is a generalization of the diagonalization method for matrices with repeated eigenvalues.

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