Fundamental matrix (linear differential equation)

Fundamental Matrix in Linear Systems

The fundamental matrix of ${\dot {x}}(t)=A(t)x(t)$ is the matrix $\Psi$ such that the n columns are linearly independent solutions of ${\dot {x}}(t)=A(t)x(t)$ .

By definition

${\dot {\Psi }}(t)=A(t)\Psi (t)$

i.e. $\Psi$ is a fundamental matrix of ${\dot {x}}(t)=A(t)x(t)$ if and olny if ${\dot {\Psi }}(t)=A(t)\Psi (t)$ and $\Psi$ is a non-singular matrix for all $t$ . ^[1]

References

^ Chi-Tsong Chen. 1998. Linear System Theory and Design (3rd ed.). Oxford University Press, Inc., New York, NY, USA.

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[1] Chi-Tsong Chen. 1998. Linear System Theory and Design (3rd ed.). Oxford University Press, Inc., New York, NY, USA.

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