An equation of the form $\mathbf {x'=Ax}$ with eigenvalue $\lambda =\alpha +\beta i$ and corresponding eigenvector $\xi ={\begin{bmatrix}\mathbf {a} \\\mathbf {c} i\end{bmatrix}}$ has the partial solution

\mathbf {x} =\gamma _{1}{\begin{bmatrix}\mathbf {a} \cos \beta t\\-\mathbf {c} \sin \beta t\end{bmatrix}}e^{\alpha t}+\gamma _{2}{\begin{bmatrix}\mathbf {a} \sin \beta t\\\mathbf {c} \cos \beta t\end{bmatrix}}e^{\alpha t}+\cdots

More generally, for a corresponding eigenvector $\xi ={\begin{bmatrix}\mathbf {a} \\\mathbf {b} +\mathbf {c} i\end{bmatrix}}$

\mathbf {x} =\gamma _{1}{\begin{bmatrix}\mathbf {a} \cos \beta t\\\mathbf {b} \cos \beta t-\mathbf {c} \sin \beta t\end{bmatrix}}e^{\alpha t}+\gamma _{2}{\begin{bmatrix}\mathbf {a} \sin \beta t\\\mathbf {c} \cos \beta t+\mathbf {b} \cos \beta t\end{bmatrix}}e^{\alpha t}+\cdots

For a complex eigenvalue, if after substitution into the original matrix and reduction to row echelon form we get a matrix such as

{\begin{bmatrix}\mathbf {a} +\mathbf {b} i&{\mbox{c}}\\0&0\end{bmatrix}}

, the eigenvectors (if we have conjugate pairs of eigenvalues) are

{\begin{bmatrix}-{\mbox{c}}\\\mathbf {a} \pm \mathbf {b} i\end{bmatrix}}

Headways[edit]

The headway between vehicles in public transit systems is the amount of time (usually in minutes) that elapses between two vehicles passing the same point traveling in the same direction on a given route. This term is most frequently applied to rail transport, where the number of tracks is limited and signaling capabilities (usually) control the headway time.

A headway, by definition, is simply a term denoting the amount of time that does elapse; it does not need to be the minimum physically possible time that must elapse between departures to prevent accidents. For example, the headway for a commuter train is usually different depending on time of day – during rush hour, headways are shorter, whereas during the off-peak hours, they are longer. A smaller headway signifies more frequent service.

Apart from practical considerations (i.e. not enough passengers to justify running very frequent service), headways are usually controlled by the speed and capabilities of the track's signaling systems, as well as the spacing of signals. Fully enclosed systems, such as metros, are usually able to achieve shorter headways due to a lack of potential "interference" from outside sources; the Moscow Metro claims headways of just 90 seconds.

Some rapid transit systems measure their capacity in train-pairs per hour. The number of train-pairs per hour is twice the number of trains that pass a given point during the course of sixty minutes. Unlike headways, a larger number of train-pairs per hour signifies more frequent service.

Train-pairs per hour can be converted into headways as follows:

Divide train-pairs by two (2) and invert the number
Multiply by 60 min/hour
Perform calculations, convert into minutes and seconds.

$x\ train{\mbox{-}}pairs/hour={\frac {2\times 1\ hr}{x\ train{\mbox{-}}pairs/hour}}={\frac {120\ min}{x\ train{\mbox{-}}pairs/hour}}={\frac {120}{x}}\ minutes\ [per\ train]$

So, for a rating of 84 train-pairs per hour, as currently used on some Moscow Metro lines,

${\frac {2\ hr}{84\ train{\mbox{-}}pairs}}={\frac {120\ min}{84\ train{\mbox{-}}pairs}}=1.42\ min/train=1\ min\ 26\ sec$