Fundamental matrix of a linear differential equation ($X^{\prime }=A(t)X,\det X\ne 0$)

For a system of linear differential equations $u^{\prime }=A(t)u$, a fundamental matrix has columns forming a basis of its solution space. It solves $X^{\prime }=A(t)X$ and is invertible at every time in its interval. Every solution is $u(t)=X(t)c$ for a constant vector $c$: differentiating $X^{-1}u$ gives zero. Invertibility at one time implies invertibility throughout the interval by uniqueness of the initial value problem. This notion is distinct from a Markov-chain fundamental matrix or the projective-geometric matrix used in stereo vision.

If $\dot{\Phi }=A(t)\Phi$ and $\Phi (0)=I$, differentiation of the determinant gives $\det \Phi (t)=\exp ({\int }_0^t\mathrm{tr}A(s)ds)$. For a planar periodic orbit this gives the nontrivial Floquet multiplier, since the tangent multiplier is one.