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Nonnegative Matrices

M. Vidyasagar

in Hidden Markov Processes: Theory and Applications to Biology

Published in print:
2014
Published Online:
October 2017
ISBN:
9780691133157
eISBN:
9781400850518
Item type:
chapter
Publisher:
Princeton University Press
DOI:
10.23943/princeton/9780691133157.003.0003
Subject:
Mathematics, Probability / Statistics

This chapter deals with nonnegative matrices, which are relevant in the study of Markov processes because the state transition matrix of such a process is a special kind of nonnegative matrix, known ... More


GROUP ORBITALLY CONSTRAINED APPROXIMATION

Moody T. Chu and Gene H. Golub

in Inverse Eigenvalue Problems: Theory, Algorithms, and Applications

Published in print:
2005
Published Online:
September 2007
ISBN:
9780198566649
eISBN:
9780191718021
Item type:
chapter
Publisher:
Oxford University Press
DOI:
10.1093/acprof:oso/9780198566649.003.0009
Subject:
Mathematics, Applied Mathematics

Group actions together with properly formulated objective functions can offer a channel to tackle various classical or new and challenging problems rising from applied linear algebra. This chapter ... More


Indefinite inner products: Conjugation

Leiba Rodman

in Topics in Quaternion Linear Algebra

Published in print:
2014
Published Online:
October 2017
ISBN:
9780691161853
eISBN:
9781400852741
Item type:
chapter
Publisher:
Princeton University Press
DOI:
10.23943/princeton/9780691161853.003.0010
Subject:
Mathematics, Algebra

This chapter studies matrices (or linear transformations) that are selfadjoint or skewadjoint with respect to a nondegenerate hermitian or skewhermitian inner product. As an application of the ... More


Mixed matrix pencils: Nonstandard involutions

Leiba Rodman

in Topics in Quaternion Linear Algebra

Published in print:
2014
Published Online:
October 2017
ISBN:
9780691161853
eISBN:
9781400852741
Item type:
chapter
Publisher:
Princeton University Press
DOI:
10.23943/princeton/9780691161853.003.0012
Subject:
Mathematics, Algebra

This chapter also studies the canonical forms of mixed quaternion matrix pencils, i.e., such that one of the two matrices is φ‎-hermitian and the other is φ‎-skewhermitian, with respect to ... More


Skewhermitian and mixed pencils

Leiba Rodman

in Topics in Quaternion Linear Algebra

Published in print:
2014
Published Online:
October 2017
ISBN:
9780691161853
eISBN:
9781400852741
Item type:
chapter
Publisher:
Princeton University Press
DOI:
10.23943/princeton/9780691161853.003.0009
Subject:
Mathematics, Algebra

This chapter turns to matrix pencils of the form A + tB, where one of the matrices A or B is skewhermitian and the other may be hermitian or skewhermitian. Canonical forms of such matrix pencils are ... More


Short Recurrences for Generating Orthogonal Krylov Subspace Bases

Jörg Liesen and Zdenek Strakos

in Krylov Subspace Methods: Principles and Analysis

Published in print:
2012
Published Online:
January 2013
ISBN:
9780199655410
eISBN:
9780191744174
Item type:
chapter
Publisher:
Oxford University Press
DOI:
10.1093/acprof:oso/9780199655410.003.0004
Subject:
Mathematics, Applied Mathematics, Algebra

This chapter links Krylov subspace methods to classical topics of linear algebra. The main goal is to explain when a Krylov sequence can be orthogonalised with an optimal (Arnoldi-type) short ... More


Indefinite inner products: Nonstandard involution

Leiba Rodman

in Topics in Quaternion Linear Algebra

Published in print:
2014
Published Online:
October 2017
ISBN:
9780691161853
eISBN:
9781400852741
Item type:
chapter
Publisher:
Princeton University Press
DOI:
10.23943/princeton/9780691161853.003.0013
Subject:
Mathematics, Algebra

This chapter fixes a nonstandard involution φ‎. It introduces indefinite inner products defined on Hn×1 of the symmetric and skewsymmetric types associated with φ‎ and matrices having symmetry ... More


Appendix: Real and complex canonical forms

Leiba Rodman

in Topics in Quaternion Linear Algebra

Published in print:
2014
Published Online:
October 2017
ISBN:
9780691161853
eISBN:
9781400852741
Item type:
chapter
Publisher:
Princeton University Press
DOI:
10.23943/princeton/9780691161853.003.0015
Subject:
Mathematics, Algebra

This chapter presents canonical forms for real and complex matrices and for pairs of real and complex matrices, or matrix pencils, with symmetries. All these forms are known, and most are well-known. ... More


Smith form and Kronecker canonical form

Leiba Rodman

in Topics in Quaternion Linear Algebra

Published in print:
2014
Published Online:
October 2017
ISBN:
9780691161853
eISBN:
9781400852741
Item type:
chapter
Publisher:
Princeton University Press
DOI:
10.23943/princeton/9780691161853.003.0007
Subject:
Mathematics, Algebra

This chapter treats matrix polynomials with quaternion coefficients. A diagonal form (known as the Smith form), which asserts that every quaternion matrix polynomial can be brought to a diagonal form ... More


Pencils of hermitian matrices

Leiba Rodman

in Topics in Quaternion Linear Algebra

Published in print:
2014
Published Online:
October 2017
ISBN:
9780691161853
eISBN:
9781400852741
Item type:
chapter
Publisher:
Princeton University Press
DOI:
10.23943/princeton/9780691161853.003.0008
Subject:
Mathematics, Algebra

This chapter is concerned with the case when both matrices A and B are hermitian. Full and detailed proofs of the canonical forms under strict equivalence and simultaneous congruence are provided, ... More


Symmetric matrices and congruence

Leiba Rodman

in Topics in Quaternion Linear Algebra

Published in print:
2014
Published Online:
October 2017
ISBN:
9780691161853
eISBN:
9781400852741
Item type:
chapter
Publisher:
Princeton University Press
DOI:
10.23943/princeton/9780691161853.003.0004
Subject:
Mathematics, Algebra

This chapter develops diagonal canonical forms and proves inertia theorems for hermitian and skewhermitian matrices with respect to involutions (including the conjugation). These canonical forms ... More


Measuring Multivariate Selection

Bruce Walsh and Michael Lynch

in Evolution and Selection of Quantitative Traits

Published in print:
2018
Published Online:
September 2018
ISBN:
9780198830870
eISBN:
9780191868986
Item type:
chapter
Publisher:
Oxford University Press
DOI:
10.1093/oso/9780198830870.003.0030
Subject:
Biology, Evolutionary Biology / Genetics, Biochemistry / Molecular Biology

This chapter extends many of the results from Chapter 29 on single trait-fitness associations to the multiple trait setting. It examines the estimate of multivariate fitness surfaces, starting with ... More


The Stroh Formalism

T. T. C. Ting

in Anisotropic Elasticity: Theory and Applications

Published in print:
1996
Published Online:
November 2020
ISBN:
9780195074475
eISBN:
9780197560280
Item type:
chapter
Publisher:
Oxford University Press
DOI:
10.1093/oso/9780195074475.003.0008
Subject:
Chemistry, Materials Chemistry

In this chapter we study Stroh's sextic formalism for two-dimensional deformations of an anisotropic elastic body. The Stroh formalism can be traced to the work of ... More


Transformation of the Elasticity Matrices and Dual Coordinate Systems

T. T. C. Ting

in Anisotropic Elasticity: Theory and Applications

Published in print:
1996
Published Online:
November 2020
ISBN:
9780195074475
eISBN:
9780197560280
Item type:
chapter
Publisher:
Oxford University Press
DOI:
10.1093/oso/9780195074475.003.0010
Subject:
Chemistry, Materials Chemistry

When the elasticity matrices are referred to a rotated coordinate system their elements change and assume different values. We will show in this chapter that, under ... More


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