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Matrix Differential Equation

A matrix differential equation is a differential equation in which the unknown is a function whose values are matrices. A first-order linear matrix differential equation has the form

X^'(t)=A(t)X(t)+F(t).

It is equivalent to a coupled system of scalar ordinary differential equations for the entries of X. When F=0, a nonsingular solution X is called a solution matrix, and every solution of the corresponding vector differential equation can then be written x(t)=X(t)c.

For a constant matrix A, the initial value problem

X^'(t)=AX(t), X(0)=I

has solution X(t)=e^(tA), given by a matrix exponential.

See also

Matrix Exponential, Ordinary Differential Equation, Vector Differential Equation

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References

Horn, R. A. and Johnson, C. R. Topics in Matrix Analysis. Cambridge, England: Cambridge University Press, 1994.

Cite this as:

Weisstein, Eric W. "Matrix Differential Equation." From MathWorld--A Wolfram Resource. https://mathworld.wolfram.com/MatrixDifferentialEquation.html

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