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

In summary, to go from eigenvalues and eigenvectors to a feasible original matrix, you can use the relationship between the original matrix A and the diagonal matrix D. If A has n independent eigenvectors, then A can be expressed as P^{-1}DP, with P being the matrix with the eigenvectors of A as columns and D being the diagonal matrix with the eigenvalues of A on the diagonal. If A is not diagonalizable, then D will be the Jordan Normal Form matrix and P will have to be supplemented with generalized eigenvectors. Fortunately, the generalized eigenvectors of A are the same as those of D.
  • #1
MikeDietrich
31
0

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!
 
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  • #2
Start with how are the original matrix A and the diagonal matrix D related and then solve for A.
 
  • #3
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, [itex]PAP^{-1}= D[/itex] where D is the diagonal matrix with the eigenvalues of A on the diagonal. From that, [itex]A= P^{-1}DP[/itex]. 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.
 

1. What is an eigenvalue?

An eigenvalue is a scalar value that represents how an eigenvector is scaled when multiplied by a particular matrix.

2. What is an eigenvector?

An eigenvector is a vector that does not change its direction when multiplied by a particular matrix. It only changes by a scalar factor known as the eigenvalue.

3. How do I find the eigenvalues of a matrix?

To find the eigenvalues of a matrix, you need to solve the characteristic equation det(A-λI) = 0, where A is the matrix, λ is the eigenvalue, and I is the identity matrix.

4. How do I find the eigenvectors of a matrix?

To find the eigenvectors of a matrix, you need to substitute the eigenvalues found in the previous step into the equation (A-λI)x = 0 and solve for x. The resulting x values are the eigenvectors.

5. How do I determine the original matrix from its eigenvalues and eigenvectors?

The original matrix can be reconstructed using the equation A = PDP^-1, where P is a matrix whose columns are the eigenvectors and D is a diagonal matrix with the eigenvalues on the diagonal. This is known as the diagonalization process.

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