derivative of inverse matrix

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derivative of inverse matrix

Theorem 1.

Suppose $A$ is a square matrix depending on a real parameter $t$ taking values in an open set $I\subseteq\mathbbmss{R}$ . Further, suppose all component functions in $A$ are differentiable, and $A(t)$ is invertible for all $t$ . Then, in $I$ , we have

\frac{dA^{{-1}}}{dt}=-A^{{-1}}\frac{dA}{dt}A^{{-1}},

where $\frac{d}{dt}$ is the derivative.

Proof.

Suppose $a_{{ij}}(t)$ are the component functions for $A$ , and $a^{{jk}}(t)$ are component functions for $A^{{-1}}(t)$ . Then for each $t$ we have

\sum_{{j=1}}^{n}a_{{ij}}(t)a^{{jk}}(t)=\delta_{i}^{k}

where $n$ is the order of $A$ , and $\delta_{i}^{k}$ is the Kronecker delta symbol. Hence

\sum_{{j=1}}^{n}\frac{da_{{ij}}}{dt}a^{{jk}}+a_{{ij}}\frac{da^{{jk}}}{dt}=0,

that is,

\frac{dA}{dt}A^{{-1}}=-A\frac{dA^{{-1}}}{dt}

from which the claim follows. ∎

Type of Math Object:

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derivative of inverse matrix

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derivative of inverse matrix

Theorem 1.

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derivative of inverse matrix

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derivative of inverse matrix

Theorem 1.

Proof.

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