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?
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.
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"?
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.
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