Fundamental matrix (linear differential equation)

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

is a matrix-valued function whose columns are linearly independent solutions of the system. Then every solution to the system can be written as , for some constant vector (written as a column vector of height n).

One can show that a matrix-valued function is a fundamental matrix of if and only if and is a non-singular matrix for all .

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