derivative of inverse matrix

Theorem 1.

Suppose A is a square matrixMathworldPlanetmath depending on a real parameter t taking values in an open set IR. Further, suppose all componentPlanetmathPlanetmathPlanetmath functionsMathworldPlanetmath in A are differentiableMathworldPlanetmathPlanetmath, and A(t) is invertiblePlanetmathPlanetmath for all t. Then, in I, we have


where ddt is the derivative.


Suppose aij(t) are the component functions for A, and ajk(t) are component functions for A-1(t). Then for each t we have


where n is the order of A, and δik is the Kronecker deltaDlmfPlanetmath symbol. Hence


that is,


from which the claim follows. ∎

Title derivative of inverse matrix
Canonical name DerivativeOfInverseMatrix
Date of creation 2013-03-22 14:43:52
Last modified on 2013-03-22 14:43:52
Owner matte (1858)
Last modified by matte (1858)
Numerical id 7
Author matte (1858)
Entry type Theorem
Classification msc 15-01