Finding eigenvalues and eigenspaces with only this info

[Dist]

Lets say I have a 3x3 matrix 'A' and one known eigenvalue 'z' and one known eigenvector 'x', but they don't "belong" to each other, as in Ax =/= zx
Is there a way of finding the other eigenvalues and eigenspaces of A using only this piece of information?
Thanks.

 

[Simon_Tyler]

I doubt it.

Just count the amount of information needed and available.
A 3x3 matrix has 9 numbers.
An eigenvalue is one number.
An eigenvector is two numbers (it's only defined upto a scale factor)
Thus in total you have 3<9 pieces of information.

Note that the full eigensystem is 3 eigenvalues and 3 eigenvectors
= 3*1 + 3*2 = 3*(1+2) = 9 pieces of information.

 

[HallsofIvy]

Even if the eigenvalue and eigenvector "belong" to each other, the other eigenvalues and eigenvectors could be anything. There will exist an infinite number of matrices having the given eigenvector and eigenvalue.

Or are you saying that you have a known matrix, one eigenvalue and eigenvector, and want to know if there is a simpler way of finding the other eigenvalues and eigenvectors than the usual "solve the characteristic equation"?

 

[Dist]

No, the matrix A is unknown as well. And I only want to find the eigenvalues and eigenspaces of A, not necessarily its eigenvectors.

 

[Simon_Tyler]

If you know the eigenspaces then you can choose a set of eigenvectors. So, your last statement is a little meaningless.

 

[HallsofIvy]

As I said before, given one eigenvalue and one eigenvector, whether they correspond or not, there exist an infinite number of matrices with those given eigenvalue and eigenvector and whatever other eigenvalues and eigenvectors you want to assign.

A single eigenvector and eigenvalue tell you nothing about other eigenvalues and eigenvectors.

 

[Dist]

Ok, I see, thanks.
But say we could choose arbitrary eigenvalues and eigenvectors that would go with the ones we already know. How would one go about doing that?

 

[Simon_Tyler]

Choose the remaining two eigenvalues. Choose the remaining two eigenvectors so that the whole set spans R^3. Assign the eigenvalues to eigenvectors. Calculate the matrix using the standard diagonalisation theorem http://mathworld.wolfram.com/MatrixDiagonalization.html