derivative of inverse matrix
Theorem 1.
Suppose A is a square matrix depending on a real parameter t
taking values in an open set I⊆R. Further, suppose all
component
functions
in A are differentiable
, and A(t) is invertible
for all t. Then, in I, we have
dA-1dt=-A-1dAdtA-1, |
where ddt is the derivative.
Proof.
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
n∑j=1aij(t)ajk(t)=δki |
where n is the order of A, and
δki is the Kronecker delta symbol. Hence
n∑j=1daijdtajk+aijdajkdt=0, |
that is,
dAdtA-1=-AdA-1dt |
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 |