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## Computing matrix symmetrizers. Part 2: new methods using eigendata and linear means; a comparison

Fonte: Elsevier
Publicador: Elsevier

Tipo: info:eu-repo/semantics/acceptedVersion; info:eu-repo/semantics/article

Publicado em 10/07/2015
ENG

Relevância na Pesquisa

37.07%

#Symmetric matrix factorization#symmetrizer#symmetrizer computation#eigenvalue method#linear equation#principal subspace computation#matrix optimization#numerical algorithm#MATLAB code#Matemáticas

Over any field F every square matrix A can be factored into the product of two symmetric matrices as A = S1 . S2 with S_i = S_i^T ∈ F^(n,n) and either factor can be chosen nonsingular, as was discovered by Frobenius in 1910. Frobenius’ symmetric matrix factorization has been lying almost dormant for a century. The first successful method for computing matrix symmetrizers, i.e., symmetric matrices S such that SA is symmetric, was inspired by an iterative linear systems algorithm of Huang and Nong (2010) in 2013 [29, 30]. The resulting iterative algorithm has solved this computational problem over R and C, but at high computational cost. This paper develops and tests another linear equations solver, as well as eigen- and principal vector or Schur Normal Form based algorithms for solving the matrix symmetrizer problem numerically. Four new eigendata based algorithms use, respectively, SVD based principal vector chain constructions, Gram-Schmidt orthogonalization techniques, the Arnoldi method, or the Schur Normal Form of A in their formulations. They are helped by Datta’s 1973 method that symmetrizes unreduced Hessenberg matrices directly. The eigendata based methods work well and quickly for generic matrices A and create well conditioned matrix symmetrizers through eigenvector dyad accumulation. But all of the eigen based methods have differing deficiencies with matrices A that have ill-conditioned or complicated eigen structures with nontrivial Jordan normal forms. Our symmetrizer studies for matrices with ill-conditioned eigensystems lead to two open problems of matrix optimization.; This research was partially supported by the Ministerio de Economía y Competitividad of Spain through the research grant MTM2012-32542.

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## Generators of algebraic covariant derivative curvature tensors and Young symmetrizers

Fonte: Universidade Cornell
Publicador: Universidade Cornell

Tipo: Artigo de Revista Científica

Publicado em 02/10/2003

Relevância na Pesquisa

26.29%

#Mathematics - Combinatorics#Computer Science - Symbolic Computation#Mathematics - Differential Geometry#53B20#15A72#05E10#16D60#05-04

We show that the space of algebraic covariant derivative curvature tensors R'
is generated by Young symmetrized tensor products W*U or U*W, where W and U are
covariant tensors of order 2 and 3 whose symmetry classes are irreducible and
characterized by the following pairs of partitions: {(2),(3)}, {(2),(2 1)} or
{(1 1),(2 1)}. Each of the partitions (2), (3) and (1 1) describes exactly one
symmetry class, whereas the partition (2 1) characterizes an infinite set S of
irreducible symmetry classes. This set S contains exactly one symmetry class
S_0 whose elements U can not play the role of generators of tensors R'. The
tensors U of all other symmetry classes from S\{S_0} can be used as generators
for tensors R'. Foundation of our investigations is a theorem of S. A. Fulling,
R. C. King, B. G. Wybourne and C. J. Cummins about a Young symmetrizer that
generates the symmetry class of algebraic covariant derivative curvature
tensors. Furthermore we apply ideals and idempotents in group rings C[Sr], the
Littlewood-Richardson rule and discrete Fourier transforms for symmetric groups
Sr. For certain symbolic calculations we used the Mathematica packages Ricci
and PERMS.; Comment: 18 pages. Chapter for a book "Progress in Computer Science Research"...

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## On the symmetry classes of the first covariant derivatives of tensor fields

Fonte: Universidade Cornell
Publicador: Universidade Cornell

Tipo: Artigo de Revista Científica

Publicado em 06/01/2003

Relevância na Pesquisa

26.29%

#Mathematics - Combinatorics#Computer Science - Symbolic Computation#Mathematics - Differential Geometry#53B20, 15A72, 05E10, 16D60, 05-04

We show that the symmetry classes of torsion-free covariant derivatives
$\nabla T$ of r-times covariant tensor fields T can be characterized by
Littlewood-Richardson products $\sigma [1]$ where $\sigma$ is a representation
of the symmetric group $S_r$ which is connected with the symmetry class of T.
If $\sigma = [\lambda]$ is irreducible then $\sigma [1]$ has a multiplicity
free reduction $[\lambda][1] = \sum [\mu]$ and all primitive idempotents
belonging to that sum can be calculated from a generating idempotent e of the
symmetry class of T by means of the irreducible characters or of a discrete
Fourier transform of $S_{r+1}$. We apply these facts to derivatives $\nabla S$,
$\nabla A$ of symmetric or alternating tensor fields. The symmetry classes of
the differences $\nabla S - sym(\nabla S)$ and $\nabla A - alt(\nabla A)$ are
characterized by Young frames (r, 1) and (2, 1^{r-1}), respectively. However,
while the symmetry class of $\nabla A - alt(\nabla A)$ can be generated by
Young symmetrizers of (2, 1^{r-1}), no Young symmetrizer of (r, 1) generates
the symmetry class of $\nabla S - sym(\nabla S)$. Furthermore we show in the
case r = 2 that $\nabla S - sym(\nabla S)$ and $\nabla A - alt(\nabla A)$ can
be applied in generator formulas of algebraic covariant derivative curvature
tensors. For certain symbolic calculations we used the Mathematica packages
Ricci and PERMS.; Comment: 21 pages. Sent in to Seminaire Lotharingien de Combinatoire:
http://www.mat.univie.ac.at/~slc/

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## Short formulas for algebraic covariant derivative curvature tensors via Algebraic Combinatorics

Fonte: Universidade Cornell
Publicador: Universidade Cornell

Tipo: Artigo de Revista Científica

Publicado em 08/12/2003

Relevância na Pesquisa

26.29%

#Mathematics - Combinatorics#Computer Science - Symbolic Computation#Mathematics - Differential Geometry#53B20, 15A72, 05E10, 16D60, 05-04

We consider generators of algebraic covariant derivative curvature tensors R'
which can be constructed by a Young symmetrization of product tensors W*U or
U*W, where W and U are covariant tensors of order 2 and 3. W is a symmetric or
alternating tensor whereas U belongs to a class of the infinite set S of
irreducible symmetry classes characterized by the partition (2,1). Using
Computer Algebra we search for such generators whose coordinate representations
are polynomials with a minimal number of summands. For a generic choice of the
symmetry class of U we obtain lengths of 16 or 20 summands if W is symmetric or
skew-symmetric, respectively. In special cases these numbers can be reduced to
the minima 12 or 10. If these minima occur then U admits an index commutation
symmetry. Furthermore minimal lengths are possible if U is formed from
torsion-free covariant derivatives of symmetric or alternating 2-tensor fields.
Foundation of our investigations is a theorem of S. A. Fulling, R. C. King, B.
G. Wybourne and C. J. Cummins about a Young symmetrizer that generates the
symmetry class of algebraic covariant derivative curvature tensors. Furthermore
we apply ideals and idempotents in group rings C[S_r] and discrete Fourier
transforms for symmetric groups S_r. For symbolic calculations we used the
Mathematica packages Ricci and PERMS.; Comment: 38 pages

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