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