From Exponentials to Logarithms

The matrix logarithm is the inverse operation to the matrix exponential. For a nonsingular matrix $A$, $L$ such that $e^L=A$ is called $log\left(A\right)$. One way to compute it for 2x2 matrices with distinct positive eigenvalues is via eigendecomposition. Write $A=V\Lambda V^{-}$ where $\Lambda$ is diagonal. Then $log\left(A\right)=Vlog\left(\Lambda \right)V^{-}$, applying the ordinary logarithm to the eigenvalues.

This calculator performs exactly that. Given entries of $A$, it finds the eigenvalues using the quadratic formula applied to the characteristic equation $\lambda 2^{}-t\lambda +d$ where $t$ is the trace $a$₁₁+a₂₂ and $d$ is the determinant $a$₁₁a₂₂-a₁₂a₂₁. Provided the eigenvalues are positive and real, it constructs $V$ from the eigenvectors, computes $log\left(\Lambda \right)$, and rebuilds the result.

Because the eigenvectors may be badly scaled, we normalize them before inversion to avoid numerical instability. If the matrix has repeated eigenvalues or complex eigenvalues, the logarithm may not be unique. This simple implementation handles only distinct positive eigenvalues, which covers many symmetric or positive-definite cases encountered in practice.

Matrix logarithms appear in differential geometry and control theory. They map a transformation back to a generator so that exponentiating the generator reproduces the original action. In computer graphics, this idea enables interpolation of rotations through the Lie algebra of the rotation group. By studying the logarithm of a matrix, we uncover how repeated infinitesimal actions combine to form the overall transformation.

After entering a 2x2 matrix, press the button to compute $log\left(A\right)$. The output displays the resulting matrix with each entry rounded to six decimal places. If the eigenvalues are not positive real numbers, the calculator warns that the method is inapplicable.