Linear Algebra meets Lie Algebra: The Kostant–Wallach theory

In two languages, Linear Algebra and Lie Algebra, we describe the results of Kostant and Wallach on the fibre of matrices with prescribed eigenvalues of all leading principal submatrices. In addition, we present a brief introduction to basic notions in Algebraic Geometry, Integrable Systems, and Lie Algebra aimed at specialists in Linear Algebra. The Lie Algebra description lets us indicate the significance of the Kostant–Wallach results based on their parameterization of square complex matrices.