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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?

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

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.