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 ( t ) {\displaystyle \Psi (t)} is a non-singular matrix for all t {\displaystyle t} . Moreover, if the entries of A ( t ) {\displaystyle A(t)} are continuous in t {\displaystyle t} , any solution to Ψ ˙ ( t ) = A ( t ) Ψ ( t ) {\displaystyle {\dot {\Psi }}(t)=A(t)\Psi (t)} which is a non-singular matrix for any single value of t {\displaystyle t} , is automatically a non-singular matrix at all other values of t {\displaystyle t} . Thus in this case, to check that Ψ {\displaystyle \Psi } is a fundamental matrix for this equation, it sufficient to check that it is non-singular at a single value of t {\displaystyle t} . Moreover, if there is at-least one choice of fundamental matrix for a given system, then for each choice of non-singular matrix B {\displaystyle B} , there is exactly one fundamental matrix solution Ψ {\displaystyle \Psi } such that Ψ ( 0 ) = B {\displaystyle \Psi (0)=B} . The same result holds if 0 {\displaystyle 0} is replaced with any fixed value t 0 {\displaystyle t_{0}} . Also, if Ψ 0 {\displaystyle \Psi _{0}} is any fundamental matrix for this equation, then for any non-singular matrix C {\displaystyle C} , the matrix Ψ ( t ) = Ψ 0 ( t ) C {\displaystyle \Psi (t)=\Psi _{0}(t)C} is also a fundamental matrix. In particular, if Ψ 0 {\displaystyle \Psi _{0}} is any fixed fundamental solution for a given equation, then all other fundamental solutions for this equation are of the form Ψ 0 ( t ) C {\displaystyle \Psi _{0}(t)C} .
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 Flow Linear differential equation Liouville's formula Systems of ordinary differential equations
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