14.4: Matrix Inversion

Given a matrix A, finding the inverse B = A⁻¹ involves solving the system of equations that satisfies

\[\mathbf{AB=BA=I}\]

with I the identity matrix. (The identity matrix is zero everywhere except at its diagonal entries, which are one.)

In the particular case of orthonormal matrices, which columns are all orthogonal to each other and of length one, the inverse is equivalent to the transpose, i.e.

\[\mathbf{A}^{-1}=\mathbf{A}^{T}\]

This is important, as rotation matrices are orthonormal. In case a matrix is not quadratic, we can calculate the pseudo-inverse, which is defined by

\[\mathbf{A}^{+}=\mathbf{A}^{T}(\mathbf{AA}^{T})^{-1}\]

and is often used in finding an inverse kinematic solution.

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