I'm just starting to read about eigenvalues. My textbook says that $A\vec{x}=\lambda\vec{x},\enspace\vec{x}\ne\vec{0}$ (the definition of eigenvalue $\lambda$) is equivalent to asking that the homogeneous system $(\lambda I-A)\vec{x}=\vec{0}$ has a non-trivial solution. Is this a simple matter of some basic matrix algebra? For example, I can see:

$A\vec{x}=\lambda\vec{x}\Rightarrow A\vec{x}-\lambda\vec{x}=\vec{0}\Rightarrow (A-\lambda)\vec{x}=\vec{0}$

but not sure how to get from $(A-\lambda)\vec{x}$ to $(\lambda I-A)\vec{x}$