Apply a matrix to Eigenvectors <=> Apply an eigenvalue to this vector
Iterative methods often depend on applying matrix
is repeatedly applied to an eigenvector , one of two things can happen.
, then will vanish as approaches infinity (Figure 9).
, then will grow to infinity (Figure 10).
-cite from An Introduction to the Conjugate Gradient Method Without the Agonizing Pain
Any vector can be written as the sum of eigenvectors if matrix
Thus, one can examine the effect of
Example: Jacobi iterations
, whose diagonal elements are identical to those of , and whose off-diagonal elements are zero; , whose whose diagonal elements are zero, and whose off-diagonal elements are identical to those of .
Suppose we start with some arbitrary vector
Spectral radius of a matrix