• Source: Fundamental matrix (linear differential equation)

In mathematics, a fundamental matrix of a system of n homogeneous linear ordinary differential equations







x

˙



(
t
)
=
A
(
t
)

x

(
t
)


{\displaystyle {\dot {\mathbf {x} }}(t)=A(t)\mathbf {x} (t)}

is a matrix-valued function



Ψ
(
t
)


{\displaystyle \Psi (t)}

whose columns are linearly independent solutions of the system.
Then every solution to the system can be written as




x

(
t
)
=
Ψ
(
t
)

c



{\displaystyle \mathbf {x} (t)=\Psi (t)\mathbf {c} }

, for some constant vector




c



{\displaystyle \mathbf {c} }

(written as a column vector of height n).

A matrix-valued function



Ψ


{\displaystyle \Psi }

is a fundamental matrix of







x

˙



(
t
)
=
A
(
t
)

x

(
t
)


{\displaystyle {\dot {\mathbf {x} }}(t)=A(t)\mathbf {x} (t)}

if and only if






Ψ
˙



(
t
)
=
A
(
t
)
Ψ
(
t
)


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

and



Ψ


{\displaystyle \Psi }

is a non-singular matrix for all



t


{\displaystyle t}

.




Control theory


The fundamental matrix is used to express the state-transition matrix, an essential component in the solution of a system of linear ordinary differential equations.




See also


Linear differential equation
Liouville's formula
Systems of ordinary differential equations




References

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