Lecture 22

November 3, 2014

Filed under: 2014 Fall — hkuyklau @ 10:23 PM

A brief summary: Given a linear operator ${T:V\rightarrow V}$ on finite-dimensional vector space ${V}$ .

For any ordered bases ${E}$ and ${F}$ ,
$\displaystyle M_{EE}(T)$ is similar to ${M_{FF}(T)}$ .
${\lambda}$ is an eigenvalue of ${T}$ ${\Leftrightarrow}$ ${\lambda}$ is an eigenvalue of (any) matrix representation of ${T}$ w.r.t an ordered basis.
Their eigenvectors are related via the coordinate isomorphism. (See p.458 in Lect21.pdf)
If ${V}$ has a ${T}$ -invariant subspace, then we are able to find a matrix representation in nice block form. (See p.455.)