The Canonical Generalized Polar Decomposition
SIAM Journal on Matrix Analysis and Applications
Symplectic lattice reduction and NTRU
EUROCRYPT'06 Proceedings of the 24th annual international conference on The Theory and Applications of Cryptographic Techniques
| Hi-index | 0.00 |
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 are given.