Chapter 15: Many interesting mathematical ideas evolve from analogies. If we think of matrices as analogs of complex numbers, then the representation z = a + bi suggests the Cartesian decomposition A = H + iK of a square complex matrix, in which Hermitian matrices play the role of real numbers. Hermitian matrices with nonnegative eigenvalues are natural analogs of nonnegative real numbers. They arise in statistics (correlation matrices and the normal equations for least-squares problems), Lagrangian mechanics (the kinetic energy functional), and quantum mechanics (density matrices). They are the subject of this chapter.

We study the solution of square systems of linear equations with a nonsingular matrices, we provide sufficient conditions for the classical Gaussian elimination with pivoting to proceed without failure. We introduce the LU factorization of a matrix, and several of its variants, like Cholesky factorization.

Chapter 3 introduces the notion of a contractive multiplier between weighted Hardy–Fock spaces (the analog of a Schur-class function for the classical setting). Unlike the classical case, in this general setting the notion of inner partitions into a number of distinct cases: (i) strictly inner (isometric multiplier) (ii) McCT (McCullough-Trent) inner (partially isometric multiplier), (iii) Bergman inner (contractive multiplier which is isometric when restricted to constants). For appropriately restricted pairs of input/output vectorial weighted Hardy–Fock spaces, analogs of the classical connections with dissipative/conservative linear input/state/output multidimensional linear systems, kernel decompositions, as well as corresponding generalized orthogonal decompositions of the ambient weighted Hardy–Fock space as a sum of a backward and a forward-shift-invariant subspace, are explored. These results are fundamental for the work of the succeeding Chapters.

Hierarchical optimalrecovery games are defined using a hierarchy of measurement functions. The sequence of optimal mixed minmax solutions is shown to be a martingale. Sparse rank-revealing representations of Gaussian fields are established.