Homogeneous linear systems: constant coefficients

Homogeneous linear systems of ODEs take the following form:

${\vec {x}}'=Ax$

Similar to the scalar case, we look for some exponential solutions:

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \vec{x}=e^{rt}\vec{\xi } }

Doing some basic differentiation and substitution from the above two equations, we get

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle r\vec{\xi}=A\vec{\xi} }

Notice that \xi is an eigenvector. The entire eigenspace is solutions. As such, we use the characteristic polynomial

$A-\lambda I$

where lambda is the eigenvalues.

There are cases where there are repeated roots.

There will be a general solution if A is diagonalizable and n eigenvectors are linearly independent and form a basis in R^n, where we get the fundamental set

$e^{r_{1}t}\xi ^{(1)},\ldots ,e^{r_{n}t}\xi ^{(n)}$

The Wronskian is then

$e^{t\sum r}|\xi ^{(1)},\xi ^{(1)}|$