Courant–Fischer theorem

The Courant–Fischer theorem provides a way to understand the size of the eigenvalues of a symmetric matrix. It states that the \(k\)-th smallest eigenvalue can be found by looking at the smallest maximum value of a certain quadratic form over all \(k\)-dimensional subspaces. Essentially, it relates eigenvalues to the extremal properties of vectors in these subspaces, allowing us to identify eigenvalues by optimizing (either minimizing or maximizing) the quadratic expression associated with the matrix. This theorem offers a systematic way to compute or estimate eigenvalues based on variational principles.