Английская Википедия:Fundamental matrix (linear differential equation)

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In mathematics, a fundamental matrix of a system of n homogeneous linear ordinary differential equations <math display="block"> \dot{\mathbf{x}}(t) = A(t) \mathbf{x}(t) </math> is a matrix-valued function <math> \Psi(t) </math> whose columns are linearly independent solutions of the system. Then every solution to the system can be written as <math>\mathbf{x}(t) = \Psi(t) \mathbf{c}</math>, for some constant vector <math>\mathbf{c}</math> (written as a column vector of height Шаблон:Mvar).

One can show that a matrix-valued function <math> \Psi </math> is a fundamental matrix of <math> \dot{\mathbf{x}}(t) = A(t) \mathbf{x}(t) </math> if and only if <math> \dot{\Psi}(t) = A(t) \Psi(t) </math> and <math> \Psi </math> is a non-singular matrix for all Шаблон:Nowrap

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

Шаблон:Matrix-stub