Know Eigenvalue and Eigenvector, How Do I Figure Out a Possible Original Matrix?

[MikeDietrich]

Homework Statement

This is a general question... I can easily go from a matrix A to its eigenvalues and then eigenvectors but how would I go from the eigenvalues and eigenvectors to a feasible original matrix?

Any thoughts appreciated!

 

[vela]

Start with how are the original matrix A and the diagonal matrix D related and then solve for A.

 

[HallsofIvy]

To expand on vela's response- presumably you know that if A has n independent eigenvectors (where A is an n by n matrix) then, with P the matrix having those eigenvectors as columns, $PAP^{-1}= D$ where D is the diagonal matrix with the eigenvalues of A on the diagonal. From that, $A= P^{-1}DP$. If you are given the eigenvalues and eigenvectors of A, you can form both P and D from that information and so find A.

If A is not diagonalizable (does not have n independent eigenvectors), then it is a little harder but the same idea- D will be the Jordan Normal Form matrix with eigenvalues along the diagonal and possibly "1"s above the diagonal. There will be fewer than n eigenvectors so you will have to supplement them with "generalized eigenvectors" to form the matrix P. Fortunately, the generalized eigenvectors of A are the same as those of D so that can be done.