Past exam of the mathematics course of the year 1 of the natural sciences course of the University of Cambridge / 2025 / Paper 2 / 18 / b / iii / Solution

Since every eigenvalue is nonzero, $S$ is an invertible matrix. Multiplying $S{\mathbf{u}}_i={\lambda }_i{\mathbf{u}}_i$ by the inverse matrix givesHence the eigenvectors are unchanged and the corresponding eigenvalues are $1/{\lambda }_i$.