05 - Eigenvalue-Polynomial Characterisation

Eigenvalue-Polynomial Characterisation

Let $A\in M_n(\mathbf{F})$

$$\lambda \text{ is an eigenvalue of }A⟺\lambda \text{ is a root of }{\chi }_A(x)⟺\lambda \text{ is a root of }{m}_A(x)$$

Proof

$$\begin{array}{rl}{\chi }_A(\lambda )=0& ⟺\det (A-\lambda I)=0\\ & ⟺A-\lambda I\text{ is singular}\\ & ⟺\exists v\ne 0:(A-\lambda I)v=0\\ & ⟺\exists v\ne 0:Av=\lambda v\\ & ⟹m_A(\lambda v)=m_A(A)v=0v=0\\ & ⟹m_A(\lambda )=0(\text{as }v\ne 0)\end{array}$$

For the converse, assume $\lambda$ is a root of $m_A$ so

$$m_A(x)=g(x)(x-\lambda )\text{ for some }g$$

By minimality of $m_A$ then $g(A)\ne 0$
Hence there exists $w\in {\mathbf{F}}^n$ such that $g(A)w\ne 0$ so

$$(A-\lambda I)(g(A)w)=m_A(A)w=0$$

So $v$ is a $\lambda -$eigenvector for $A$