Fundamental matrix (linear differential equation)

In mathematics, a fundamental matrix of a system of n homogeneous linear ordinary differential equations$${\displaystyle {\dot {\mathbf {x} }}(t)=A(t)\mathbf {x} (t)}$${\displaystyle {\dot {\mathbf {x} }}(t)=A(t)\mathbf {x} (t)}is a matrix-valued function ${\displaystyle \Psi (t)}${\displaystyle \Psi (t)} whose columns are linearly independent solutions of the system.^[1] Then every solution to the system can be written as ${\displaystyle \mathbf {x} (t)=\Psi (t)\mathbf {c} }${\displaystyle \mathbf {x} (t)=\Psi (t)\mathbf {c} }, for some constant vector ${\displaystyle \mathbf {c} }${\displaystyle \mathbf {c} } (written as a column vector of height n).

A matrix-valued function ${\displaystyle \Psi }${\displaystyle \Psi } is a fundamental matrix of ${\displaystyle {\dot {\mathbf {x} }}(t)=A(t)\mathbf {x} (t)}${\displaystyle {\dot {\mathbf {x} }}(t)=A(t)\mathbf {x} (t)} if and only if ${\displaystyle {\dot {\Psi }}(t)=A(t)\Psi (t)}${\displaystyle {\dot {\Psi }}(t)=A(t)\Psi (t)} and ${\displaystyle \Psi }${\displaystyle \Psi } is a non-singular matrix for all ${\displaystyle t}${\displaystyle t}.^[2]

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.^[3]

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

• Matrices
• Differential calculus
• Ordinary differential equations
• Matrix stubs

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