Structured Factorizations in Scalar Product Spaces

Abstract

Abstract. Let A belong to an automorphism group, Lie algebra or Jordan algebra of a scalar product. When A is factored, to what extent do the factors inherit structure from A? We answer this question for the principal matrix square root, the matrix sign decomposition, and the polar decomposition. For general A, we give a simple derivation and characterization of a particular generalized polar decomposition, and we relate it to other such decompositions in the literature. Finally, we study eigendecompositions and structured singular value decompositions, considering in particular the structure in eigenvalues, eigenvectors and singular values that persists across a wide range of scalar products.A key feature of our analysis is the identification of two particular classes of scalar products, termed unitary and orthosymmetric, which serve to unify assumptions for the existence of structured factorizations. A variety of different characterizations of these scalar product classes is given.Key words. automorphism group, Lie group, Lie algebra, Jordan algebra, bilinear form, sesquilinear form, scalar product, indefinite inner product, orthosymmetric, adjoint, factorization, symplectic, Hamiltonian, pseudo-orthogonal, polar decomposition, matrix sign function, matrix square root, generalized polar decomposition, eigenvalues, eigenvectors, singular values, structure preservation.AMS subject classifications. 15A18, 15A21, 15A23, 15A57, 15A631. Introduction. The factorization of a general matrix into a product of structured factors plays a key role in theoretical and computational linear algebra. In this work we address the following question: if we apply one of the standard factorizations to a matrix that is already structured, to what extent do the factors have additional structure related to that of the original matrix?Many applications generate structure and there are potential benefits to be gained by exploiting it when developing theory and deriving algorithms. For example, algorithms that preserve structure may have reduced storage requirements and operation counts, may be more accurate, and may also provide solutions that are more physically meaningful in the presence of rounding and truncation errors.The structured matrices we consider belong to the automorphism group G, the Lie algebra L, and the Jordan algebra J associated with a scalar product, that is, a nondegenerate bilinear or sesquilinear form on K n . These classes of matrices include linear structures such as complex symmetric, pseudo-symmetric, and Hamiltonian matrices, as well as nonlinear structures such as complex orthogonal, pseudo-orthogonal and symplectic matrices. Section 2 introduces concepts and notation needed for our unified treatment of structured factorizations in scalar product spaces. We introduce two important special types of scalar product, termed unitary and orthosymmetric, and describe several equivalent ways to characterize them. The proofs of these equivalences have been delegated to an appendix to avoid disrupting the flow of the p...

The hyperbolic Schur decomposition

“…Eigenvalues of J-unitary and J-Hermitian matrices have well researched properties, nicely presented in [16,Section 7] with J-unitary matrices being referred to as members of the automorphism group G and J-Hermitian matrices being referred to as members of the Jordan algebra J. These results apply for various scalar products, but when it comes to hyperbolic products and a hyperbolic Schur decomposition, more can be said about the atomic blocks in T .…”

“…Here, we see that λ ∈ R (which also follows straight from [16,Theorem 7.6]) and α = α, i.e., α is real, so […”

“…Trivially, this means that all blocks T k of order 1 are real. Notice that if T k of order 2 has non-real eigenvalues λ 2 , due to [16,Theorem 7.6], so their partial multiplicity must be 1. If they are real, their partial multiplicity is 1 or 2, depending on the diagonalizability of T k .…”

“…It is used, for example, in the theory of relativity and in the research of the polarized light. More on the applications of such products can be found in [10,13,14,17].The Euclidean matrix decompositions have some nice structure preserving properties even in nonstandard scalar products, as shown by Mackey, Mackey and Tisseur [16], but it is often worth looking into versions of such decompositions that respect the structures related with the given scalar product. There is plenty of research on the subject, i.e., hyperbolic SVD [17,24], J 1 J 2 -SVD [9], two-sided hyperbolic SVD [20], hyperbolic CS decomposition [8,10] and indefinite QR factorization [19].…”

Orthosymmetric block rotations

“…Introduction. Orthosymmetric scalar products, introduced in [9] by Mackey, Mackey and Tisseur, still do not belong to the standard vocabulary of numerical linear algebra, even though they provide a unified setting for many modern structure preserving matrix tools (see for example [2,8,9,10]). …”