FUNDAMENTAL MATRICES and
REPEATED EIGENVALUES

The General Problem:

${\bf x}' = P(t){\bf x}$, given ${\bf x}(t_0)$.

• If ${\bf x}^{(1)}(t),\ {\bf x}^{(2)}(t),\ \ldots,\ {\bf x}^{(n)}(t)$ is a fundamental set of solutions to ${\bf x}' = P(t){\bf x}$, the fundamental matrix is $\Psi(t) = [{\bf x}^{(1)}(t)\ {\bf x}^{(2)}(t)\ \ldots\ {\bf x}^{(n)}(t)] .$
• General solution is ${\bf x}(t)=\Psi(t){\bf c}$, so ${\bf c}=\Psi^{-1}(t_0){\bf x}(t_0)$.
• The final solution is
  
  $${\bf x}(t) = \Psi(t)\Psi^{-1}(t_0){\bf x}(t_0) = \Phi(t){\bf x}(t_0) ,$$
  
  
  where $\Phi(t) = \Psi(t)\Psi^{-1}(t_0);$ note $\Phi(t_0) = I$.

Constant Coefficient Problem

${\bf x}' = A{\bf x}$, given ${\bf x}(t_0)$.

• If A has a full set of linearly independent eigenvectors, ${\bf v}^{(1)}$, ${\bf v}^{(2)}$, ..., ${\bf v}^{(n)}$, let $T = [{\bf v}^{(1)}\ {\bf v}^{(2)}\ \ldots\ {\bf v}^{(n)}] .$
• Then T diagonalizes A: T^-1AT = D, where D is a diagonal matrix with i^th diagonal entry r_i.
• If Q(t) is a diagonal matrix with e^r_it as the i^th diagonal entry, then the fundamental matrix $\Psi(t) = TQ(t)$.
• Therefore
  
  $${\bf x}(t) = \Psi(t)\Psi^{-1}(t_0){\bf x}(t_0) = TQ(t)Q^{-1}(t_0)T^{-1}{\bf x}(t_0) ;$$
  
  
  note $\Phi(t) = TQ(t)Q^{-1}(t_0)T^{-1}.$

The Matrix Exponential:

e^At.

• Series definition $e^{At}=I+ \displaystyle\sum_{n=1}^\infty \frac{A^nt^n}{n!} .$
• Solution to ${\bf x}' = A{\bf x}$, given ${\bf x}(0)$ is ${\bf x}(t) = e^{At}{\bf x}(0) .$
• Therefore $e^{At} = TQ(t)T^{-1} = \Phi(t).$

Repeated Eigenvalues

• Assume a real eigenvalue r has multiplicity k.
• If there are k independent eigenvectors ${\bf v}^{(1)}, \ldots, {\bf v}^{(n)}$,
  then the vectors ${\bf x}^{(1)}={\bf v}^{(1)}e^{rt},\ \ldots,\ {\bf x}^{(n)}={\bf v}^{(n)}e^{rt}$
  form a fundamental set of solutions.
• If there are less than k independent eigenvectors,
  then there are many cases to consider.
• Example: suppose k=2 with only one eigenvector ${\bf v}$;
  ${\bf x}^{(1)} = {\bf v}e^{rt}$, ${\bf x}^{(2)} = {\bf v}te^{rt} + {\bf w}e^{rt}$, with $(A-rI){\bf w}= {\bf v}$.
• Example: suppose k=3 with only one eigenvector ${\bf v}$;
  ${\bf x}^{(1)} = {\bf v}e^{rt}$, ${\bf x}^{(2)} = {\bf v}te^{rt} + {\bf w}e^{rt}$, with $(A-rI){\bf w}= {\bf v}$,
  and ${\bf x}^{(3)} = {\bf v}\frac{t^2}{2}e^{rt} + {\bf w}te^{rt} + {\bf z}e^{rt}$, with $(A-rI){\bf z}= {\bf w}$.