turkmath.org

Originating in its simplest form in continuum mechanics and having obvious generalizations in Hilbert spaces, the Cauchy Polar Decomposition stipulates that a non-singular matrix can be uniquely expressed as the product of an orthogonal and a symmetric positive definite matrix. It has been recently observed that with respect to the natural inner product structure on the vector space of square matrices, the orthogonal part is the element of the group of orthogonal matrices closest to the given matrix. The speaker notes that the symmetric, positive definite part can be regarded as the Lagrange multipliers in the same variational setting.