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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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