Invertible Matrix - Derivative of The Matrix Inverse

Derivative of The Matrix Inverse

Suppose that the invertible matrix A depends on a parameter t. Then the derivative of the inverse of A with respect to t is given by

To derive the above expression for the derivative of the inverse of A, one can differentiate the definition of the matrix inverse and then solve for the inverse of A:

\frac{\mathrm{d}\mathbf{A}^{-1}\mathbf{A}}{\mathrm{d}t} =\frac{\mathrm{d}\mathbf{A}^{-1}}{\mathrm{d}t}\mathbf{A} +\mathbf{A}^{-1}\frac{\mathrm{d}\mathbf{A}}{\mathrm{d}t} =\frac{\mathrm{d}\mathbf{I}}{\mathrm{d}t} =\mathbf{0}.

Subtracting from both sides of the above and multiplying on the right by gives the correct expression for the derivative of the inverse:

Similarly, if is a small number then

\left(\mathbf{A} + \epsilon\mathbf{X}\right)^{-1} = \mathbf{A}^{-1} - \epsilon \mathbf{A}^{-1} \mathbf{X} \mathbf{A}^{-1} + \mathcal{O}(\epsilon^2)\,.

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