Fundamental matrix (linear differential equation)

{{Short description|Matrix consisting of linearly independent solutions to a linear differential equation}}

{{dablink|For other senses of the term, see Fundamental matrix (disambiguation).}}

In mathematics, a fundamental matrix of a system of n homogeneous linear ordinary differential equations$$\backslash dot\{\backslash mathbf\{x\}\}(t)\; =\; A(t)\; \backslash mathbf\{x\}(t)$$is a matrix-valued function $\backslash Psi(t)$ whose columns are linearly independent solutions of the system.{{cite book |first=D. |last=Somasundaram |title=Ordinary Differential Equations: A First Course |location=Pangbourne |publisher=Alpha Science |year=2001 |isbn=1-84265-069-6 |pages=233–240 |chapter=Fundamental Matrix and Its Properties |chapter-url=https://books.google.com/books?id=PduY2CjJ1zEC&pg=PA233 }}

Then every solution to the system can be written as $\backslash mathbf\{x\}(t)\; =\; \backslash Psi(t)\; \backslash mathbf\{c\}$, for some constant vector $\backslash mathbf\{c\}$ (written as a column vector of height {{mvar|n}}).

A matrix-valued function $\backslash Psi$ is a fundamental matrix of $\backslash dot\{\backslash mathbf\{x\}\}(t)\; =\; A(t)\; \backslash mathbf\{x\}(t)$ if and only if $\backslash dot\{\backslash Psi\}(t)\; =\; A(t)\; \backslash Psi(t)$ and $\backslash Psi$ is a non-singular matrix for all {{nowrap|$t$.}}{{cite book |author=Chi-Tsong Chen |year=1998 |title=Linear System Theory and Design |edition=3rd |publisher=Oxford University Press |location=New York |isbn=0-19-511777-8 }}