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

\dot(t) = A(t) \mathbf(t)

is a matrix-valued function

\Psi(t)

whose columns are

linearly independent In the theory of vector spaces, a set of vectors is said to be if there is a nontrivial linear combination of the vectors that equals the zero vector. If no such linear combination exists, then the vectors are said to be . These concepts are ...

solutions of the system. Then every solution to the system can be written as

\mathbf(t) = \Psi(t) \mathbf

, for some constant vector

\mathbf

(written as a column vector of height ). One can show that a matrix-valued function

\Psi

is a fundamental matrix of

\dot(t) = A(t) \mathbf(t)

if and only if

\dot(t) = A(t) \Psi(t)

and

\Psi

is a

non-singular matrix In linear algebra, an -by- square matrix is called invertible (also nonsingular or nondegenerate), if there exists an -by- square matrix such that :\mathbf = \mathbf = \mathbf_n \ where denotes the -by- identity matrix and the multiplicati ...

for all

Control theory

The fundamental matrix is used to express the

state-transition matrix In control theory, the state-transition matrix is a matrix whose product with the state vector x at an initial time t_0 gives x at a later time t. The state-transition matrix can be used to obtain the general solution of linear dynamical systems ...

, an essential component in the solution of a system of linear ordinary differential equations.

References

{{Matrix classes Matrices Differential calculus *

Control theory

See also

References