Understanding Eigenspaces and Eigenvectors

[flyingpig]

Homework Statement

Suppose $$A\boldsymbol{x} = \lambda\boldsymbol{x}$$ and $$A\boldsymbol{x} - \lambda\boldsymbol{x} = \boldsymbol{0}$$

Then the $$\boldsymbol{x}$$ (vectors) that form the eigenspace are the linearly independent set of eigenvectors assuming $$A\boldsymbol{x} - \lambda\boldsymbol{x} = \boldsymbol{0}$$
has a nontrivial solution.

The Attempt at a Solution

It's true right? It's just solving the nullspace and then naming the new solutions as eigenvectors.

 

[lanedance]

yes

solving $det(A-\lambda I)$ will give you the allowable values for lambda (eigenvalues of A)

yes

then for a given $\lambda$, solving $Ax = \lambda x$ for non-trivial x, will give you the corresponding eigenvectors for that eigenvalue