Chapter 6.5, Problem 23P

Assume that the real $n\times n$ matrix $\text{A}$ has $n$ linearly independent eigenvectors ${\text{v}}_{\text{1}}\text{,}\mathrm{......}{\text{,v}}_{\text{n}}$ corresponding to the (possibly repeated and possibly complex) eigenvalues ${\text{λ}}_{\text{1}}\text{,}\mathrm{.....}{\text{,λ}}_{\text{n}}$. If $\text{T}$ is the matrix whose columns are the eigenvectors of $\text{A}$,

$\text{T=}\left[{\text{v}}_{\text{1}}\text{,}\mathrm{.......}{\text{,v}}_{\text{n}}\right],$

it is shown in Appendix A.4 that

${\text{T}}^{\text{-1}}\text{AT=D=}\left(\begin{array}{cccc}{\text{λ}}_1& 0& \mathrm{...}& 0\\ 0& {\text{λ}}_{\text{2}}& \mathrm{...}& 0\\ ⋮& & & ⋮\\ 0& 0& \mathrm{...}& {\text{λ}}_{\text{n}}\end{array}\right)$,         (i)

that is, $\text{A}$ is similar to the diagonal matrix $\text{D=diag}\left({\text{λ}}_{\text{1}}\text{,}\mathrm{......}{\text{,λ}}_{\text{n}}\right)$.

(a) Use the relation in Eq.(i) to show that

${\text{A''=TD''T}}^{\text{-1}}$ for each $n=0,1,2,\mathrm{.......}$.

(b) Use the results in part (a) Eq.(17) in definition 6.5.1, and Eq.(19) to show that

$e^{\text{At}}={\text{Te}}^{\text{Dt}}{\text{T}}^{\text{-1}}$.