Fundamental matrix (linear differential equation)
In mathematics, a fundamental matrix of a system of n homogeneous linear ordinary differential equationsis 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 .
A matrix-valued function is a fundamental matrix of if and only if and is a non-singular matrix for all Moreover, if the entries of are continuous in, any solution to which is a non-singular matrix for any single value of, is automatically a non-singular matrix at all other values of. Thus in this case, to check that is a fundamental matrix for this equation, it sufficient to check that it is non-singular at a single value of.
Moreover, if there is at-least one choice of fundamental matrix for a given system, then for each choice of non-singular matrix, there is exactly one fundamental matrix solution such that. The same result holds if is replaced with any fixed value. Also, if is any fundamental matrix for this equation, then for any non-singular matrix, the matrix is also a fundamental matrix. In particular, if is any fixed fundamental solution for a given equation, then all other fundamental solutions for this equation are of the form.