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Infinitesimal Matrix Change


Let B, A, and e be square matrices with e small, and define

 B=A(I+e),
(1)

where I is the identity matrix. Then the inverse of B is approximately

 B^(-1)=(I-e)A^(-1).
(2)

This can be seen by multiplying

BB^(-1)=(A+Ae)(A^(-1)-eA^(-1))
(3)
=AA^(-1)-AeA^(-1)+AeA^(-1)-Ae^2A^(-1)
(4)
=I-Ae^2A^(-1)
(5)
 approx I.
(6)

Note that if we instead let B^'=A+e, and look for an inverse of the form B^('-1)=A^(-1)+C, we obtain

B^'B^('-1)=(A+e)(A^(-1)+C)
(7)
=AA^(-1)+AC+eA^(-1)+eC
(8)
=I+AC+e(C+A^(-1))
(9)
 approx I.
(10)

In order to eliminate the e term, we require C=-A^(-1). However, then AC=-I, so BB^(-1)=0 so there can be no inverse of this form.

The exact inverse of B^' can be found as follows.

 B^'=A(I+e)=A(I+A^(-1)e),
(11)

so

 B^('-1)=[A(I+A^(-1)e)]^(-1).
(12)

Using a general matrix inverse identity then gives

 B^('-1)=(I+A^(-1)e)^(-1)A^(-1).
(13)

See also

Infinitesimal Rotation

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Cite this as:

Weisstein, Eric W. "Infinitesimal Matrix Change." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/InfinitesimalMatrixChange.html

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