Eigenvalue, trace, determinant and rank of a matrix 간의 관계들

Trace

• $tr(A)=\sum _{i=0}^n{\lambda }_i$

Determinant

• $det(A)=\prod _{i=0}^n{\lambda }_i$
• Proof(맞는진 모름):
  • $$det(A-\lambda I)=(-1{)}^n(\lambda -{\lambda }_1)...(\lambda -{\lambda }_n)=({\lambda }_1-\lambda )...({\lambda }_n-\lambda )$$
  • $$det(A-\lambda I){|}_{\lambda =0}=det(A)={\lambda }_1...{\lambda }_n$$

Rank

• $rank(A)=numberofnonzeroeigenvalues.$

$\lambda$가 eigenvalue of $A$ 이면 $f(\lambda )$도 $f(A)$의 eigenvalue이다.

증명:f를 테일러전개하면 됨.