# CG -- Eigenvalues (Eigenvectors) and convergency

### Apply a matrix to Eigenvectors <=> Apply an eigenvalue to this vector

Iterative methods often depend on applying matrix

When

is repeatedly applied to an eigenvector , one of two things can happen.

- If
, then will vanish as approaches infinity (Figure 9). - If
, 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 is symmetric

If

Thus, one can examine the effect of

### Example: Jacobi iterations

The matrix

, 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 .

That is

Suppose we start with some arbitrary vector

###### Spectral radius of a matrix

Thus, if