MATH319 Slides 111 Growth of exponentials of diagonable matrices 113 Conclusion of Proof

112 Exponentials of diagonable matrices

Proof. (i) The $\{X_{1},\dots,X_{n}\}$ give a basis for the space ${\bf C}^{n\times 1}$ ; see MATH220, Theorem 4.51. Also $X(t)=\exp(tA)X_{0}$ satisfies the ODE.

(ii), (iii) As usual, we introduce an invertible matrix $S=[X_{1}\,X_{2}\,\dots\,X_{n}]$ with columns the eigenvectors of $A$ such that

\exp(tA)=S\begin{bmatrix}e^{t\lambda_{1}}&0&\dots\\ 0&\ddots&0\\ 0&\dots&e^{t\lambda_{n}}\end{bmatrix}S^{-1}

\begin{bmatrix}e^{t\lambda_{1}}&0&\dots\\ 0&\ddots&0\\ 0&\dots&e^{t\lambda_{n}}\end{bmatrix}\begin{bmatrix}z_{1}\\ z_{2}\\ \vdots\\ z_{n}\end{bmatrix}=\begin{bmatrix}e^{t\lambda_{1}}z_{1}\\ e^{t\lambda_{2}}z_{2}\\ \vdots\\ e^{t\lambda_{n}}z_{n}\end{bmatrix}