On fast multiplication of a matrix by its transpose

We present a non-commutative algorithm for the multiplication of a 2x2-block-matrix by its transpose using 5 block products (3 recursive calls and 2 general products) over C or any finite field.We use geometric considerations on the space of bilinear forms describing 2x2 matrix products to obtain this algorithm and we show how to reduce the number of involved additions.The resulting algorithm for arbitrary dimensions is a reduction of multiplication of a matrix by its transpose to general matrix product, improving by a constant factor previously known reductions.Finally we propose schedules with low memory footprint that support a fast and memory efficient practical implementation over a finite field.To conclude, we show how to use our result in LDLT factorization.Comment: ISSAC 2020, Jul 2020, Kalamata, Greec

On fast multiplication of a matrix by its transpose

We present a non-commutative algorithm for the multiplication of a block-matrix by its transpose over C or any finite field using 5 recursive products. We use geometric considerations on the space of bilinear forms describing 2×2 matrix products to obtain this algorithm and we show how to reduce the number of involved additions. The resulting algorithm for arbitrary dimensions is a reduction of multiplication of a matrix by its transpose to general matrix product, improving by a constant factor previously known reductions. Finally we propose space and time efficient schedules that enable us to provide fast practical implementations for higher-dimensional matrix products

Implicit QR algorithms for palindromic and even eigenvalue problems

In the spirit of the Hamiltonian QR algorithm and other bidirectional chasing algorithms, a structure-preserving variant of the implicit QR algorithm for palindromic eigenvalue problems is proposed. This new palindromic QR algorithm is strongly backward stable and requires less operations than the standard QZ algorithm, but is restricted to matrix classes where a preliminary reduction to structured Hessenberg form can be performed. By an extension of the implicit Q theorem, the palindromic QR algorithm is shown to be equivalent to a previously developed explicit version. Also, the classical convergence theory for the QR algorithm can be extended to prove local quadratic convergence. We briefly demonstrate how even eigenvalue problems can be addressed by similar techniques. © 2008 Springer Science+Business Media, LLC

``Batman decomposition of a symmetric indefinite matrix

V práci nejprve zopakujeme vybrané základních pojmy, zejména vlastní čísla symetrických(obecně indefinitních) matic a kvadratické formy. Pak se zaměříme na vybrané zajímavé podprostory týkající se symetrických matic. Konkrétně tzv. nulový prostor, neutrální podprostor, nezáporný a nekladný podprostor. V práci ukážeme, že ne všechny tyto podprostory jsou dány jednoznačně, ale vždy je umíme zvolit tak, že např. první tři zmiňované jsou postupně svými podprostory. Toho využijeme k volbě vhodné ortonormální báze těchto podprostorů a jejich ortogonálních doplňků. Nakonec ukážeme, že takto zkonstruovaná báze (po drobných úpravách), resp. ortogonální matice, která má tyto bázové vektory jako sloupce, transformuje původní symetrickou matici na tzv. dolní blokově antitrojúhelníkový tvar. Odpovídající rozklad nazýváme Batman decomposition.At first we recapitulate some basic concepts such as eigenvalues of symmetric (in general indefinite) matrices and quadratic forms. Then, we focuse mainly on selected interesting subspaces related to symmetric matrices. Specifically, the so-called null-space, neutral subspace, nonnegative, and nonpositive subspaces. In the thesis we show that not all of these subspaces are given uniquely, in general, but we are always able to choose them in such a way that, e.g. the first three of above mentioned spaces are nested. We will use that for a choice of suitable orthonormal basis of these subspaces and their orthogonal complements. Finally, we show that a basis constructed like that (after small modifications), more precisely the orthogonal matrix having those basis vectors as columns, transforms original symmetric matrix into so-called lower block antitriangular form. We call the corresponding decomposition the Batman decomposition