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The Fundamental Theorem of Linear Algebra

Gidon Eshel

in Spatiotemporal Data Analysis

Published in print:
2011
Published Online:
October 2017
ISBN:
9780691128917
eISBN:
9781400840632
Item type:
chapter
Publisher:
Princeton University Press
DOI:
10.23943/princeton/9780691128917.003.0010
Subject:
Environmental Science, Environmental Studies

This chapter summarizes pictorially some of the linear algebraic foundations discussed thus far by revisiting the fundamental theorem of linear algebra, the unifying view of matrices, vectors, and ... 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


Matrix Properties, Fundamental Spaces, Orthogonality

Gidon Eshel

in Spatiotemporal Data Analysis

Published in print:
2011
Published Online:
October 2017
ISBN:
9780691128917
eISBN:
9781400840632
Item type:
chapter
Publisher:
Princeton University Press
DOI:
10.23943/princeton/9780691128917.003.0003
Subject:
Environmental Science, Environmental Studies

This chapter provides an introduction to linear algebra. Topics covered include vector spaces, matrix rank, fundamental spaces associated with A ɛ ℝM×N, and Gram–Schmidt orthogonalization. In ... More


Regression and Least Squares

Gidon Eshel

in Spatiotemporal Data Analysis

Published in print:
2011
Published Online:
October 2017
ISBN:
9780691128917
eISBN:
9781400840632
Item type:
chapter
Publisher:
Princeton University Press
DOI:
10.23943/princeton/9780691128917.003.0009
Subject:
Environmental Science, Environmental Studies

This chapter focuses on linear regression, the process of identifying the unique model that best explains a set of observed data among a specified class of general models. Regression thus occupies a ... More


Notation and Basic Operations

Gidon Eshel

in Spatiotemporal Data Analysis

Published in print:
2011
Published Online:
October 2017
ISBN:
9780691128917
eISBN:
9781400840632
Item type:
chapter
Publisher:
Princeton University Press
DOI:
10.23943/princeton/9780691128917.003.0002
Subject:
Environmental Science, Environmental Studies

This chapter introduces some of the basic players of the algebraic drama about to unfold, and the uniform notations used in this book. While Chapter 3 is a more formal introduction to linear algebra, ... More


The Gray World of Practical Data Analysis: An Introduction to Part 2

Gidon Eshel

in Spatiotemporal Data Analysis

Published in print:
2011
Published Online:
October 2017
ISBN:
9780691128917
eISBN:
9781400840632
Item type:
chapter
Publisher:
Princeton University Press
DOI:
10.23943/princeton/9780691128917.003.0006
Subject:
Environmental Science, Environmental Studies

This chapter provides an overview of the second part of the book. This part is the crux of the matter: how to analyze actual data. While this part builds on Part 1, especially on linear algebra ... More


Spatiotemporal Data Analysis

Gidon Eshel

Published in print:
2011
Published Online:
October 2017
ISBN:
9780691128917
eISBN:
9781400840632
Item type:
book
Publisher:
Princeton University Press
DOI:
10.23943/princeton/9780691128917.001.0001
Subject:
Environmental Science, Environmental Studies

A severe thunderstorm morphs into a tornado that cuts a swath of destruction through Oklahoma. How do we study the storm’s mutation into a deadly twister? Avian flu cases are reported in China. How ... More


Topics in Quaternion Linear Algebra

Leiba Rodman

Published in print:
2014
Published Online:
October 2017
ISBN:
9780691161853
eISBN:
9781400852741
Item type:
book
Publisher:
Princeton University Press
DOI:
10.23943/princeton/9780691161853.001.0001
Subject:
Mathematics, Algebra

Quaternions are a number system that has become increasingly useful for representing the rotations of objects in three-dimensional space and has important applications in theoretical and applied ... More


Matrix algorithms and graph partitioning: A discussion of network algorithms that use matrix and linear algebra methods, including algorithms for partitioning network nodes into groups

M. E. J. Newman

in Networks: An Introduction

Published in print:
2010
Published Online:
September 2010
ISBN:
9780199206650
eISBN:
9780191594175
Item type:
chapter
Publisher:
Oxford University Press
DOI:
10.1093/acprof:oso/9780199206650.003.0011
Subject:
Physics, Theoretical, Computational, and Statistical Physics

The preceding chapter discussed a variety of computer algorithms for calculating quantities of interest on networks, including degrees, centralities, shortest paths, and connectivity. This chapter ... More


Mixed Hodge Structures

Fouad El Zein and Lˆe D˜ung Tr ´ang

Eduardo Cattani, Fouad El Zein, Phillip A. Griffiths, and Lê Dũng Tráng (eds)

in Hodge Theory (MN-49)

Published in print:
2014
Published Online:
October 2017
ISBN:
9780691161341
eISBN:
9781400851478
Item type:
chapter
Publisher:
Princeton University Press
DOI:
10.23943/princeton/9780691161341.003.0003
Subject:
Mathematics, Geometry / Topology

This chapter discusses mixed Hodge structures (MHS). It first defines the abstract category of Hodge structures and introduces spectral sequences. The decomposition on the cohomology of Kähler ... More


Introduction

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/9780691161853.003.0001
Subject:
Mathematics, Algebra

This chapter provides a brief overview of this volume, showing that this book can be used for a variety of ways and can be used to assist the reader toward a better understanding of quaternion linear ... More


Matrix Completions, Moments, and Sums of Hermitian Squares

Mihály Bakonyi and Hugo J. Woerdeman

Published in print:
2011
Published Online:
October 2017
ISBN:
9780691128894
eISBN:
9781400840595
Item type:
book
Publisher:
Princeton University Press
DOI:
10.23943/princeton/9780691128894.001.0001
Subject:
Mathematics, Computational Mathematics / Optimization

Intensive research in matrix completions, moments, and sums of Hermitian squares has yielded a multitude of results in recent decades. This book provides a comprehensive account of this quickly ... More


Preliminaries

Jayme Vaz and Roldão da Rocha

in An Introduction to Clifford Algebras and Spinors

Published in print:
2016
Published Online:
August 2016
ISBN:
9780198782926
eISBN:
9780191826146
Item type:
chapter
Publisher:
Oxford University Press
DOI:
10.1093/acprof:oso/9780198782926.003.0001
Subject:
Physics, Theoretical, Computational, and Statistical Physics

In this chapter, a review of the main concepts, examples, and results from linear algebra are presented, as these are essential for developing the subsequent ideas in this book. As the concept of ... More


Suggested Homework

Gidon Eshel

in Spatiotemporal Data Analysis

Published in print:
2011
Published Online:
October 2017
ISBN:
9780691128917
eISBN:
9781400840632
Item type:
chapter
Publisher:
Princeton University Press
DOI:
10.23943/princeton/9780691128917.003.0013
Subject:
Environmental Science, Environmental Studies

This chapter presents some homework exercises on the topics discussed in chapters 3–8, along with their corresponding solutions. It also provides a suggested midterm exam and a final exam.


The Mathematics of Various Entertaining Subjects: Research in Games, Graphs, Counting, and Complexity, Volume 2

Jason Rosenhouse and Jennifer Beineke (eds)

Published in print:
2017
Published Online:
May 2018
ISBN:
9780691171920
eISBN:
9781400889136
Item type:
book
Publisher:
Princeton University Press
DOI:
10.23943/princeton/9780691171920.001.0001
Subject:
Mathematics, History of Mathematics

The history of mathematics is filled with major breakthroughs resulting from solutions to recreational problems. Problems of interest to gamblers led to the modern theory of probability, for example, ... More


Linear algebra

Hans Ringström

in On the Topology and Future Stability of the Universe

Published in print:
2013
Published Online:
September 2013
ISBN:
9780199680290
eISBN:
9780191760235
Item type:
chapter
Publisher:
Oxford University Press
DOI:
10.1093/acprof:oso/9780199680290.003.0012
Subject:
Mathematics, Mathematical Physics, Geometry / Topology

In kinetic theory, the matter is represented by a distribution function. In the relativistic setting, this function is defined on the mass shell, consisting of the future directed unit timelike ... More


The p-adic Simpson Correspondence (AM-193)

Ahmed Abbes, Michel Gros, and Takeshi Tsuji

Published in print:
2016
Published Online:
October 2017
ISBN:
9780691170282
eISBN:
9781400881239
Item type:
book
Publisher:
Princeton University Press
DOI:
10.23943/princeton/9780691170282.001.0001
Subject:
Mathematics, Algebra

The p-adic Simpson correspondence, recently initiated by Gerd Faltings, aims at describing all p-adic representations of the fundamental group of a proper smooth variety over a p-adic field in terms ... More


Matrices and Lie groups

Clifford Henry Taubes

in Differential Geometry: Bundles, Connections, Metrics and Curvature

Published in print:
2011
Published Online:
December 2013
ISBN:
9780199605880
eISBN:
9780191774911
Item type:
chapter
Publisher:
Oxford University Press
DOI:
10.1093/acprof:oso/9780199605880.003.0002
Subject:
Mathematics, Geometry / Topology, Mathematical Physics

This chapter discusses the following: the general linear group; Lie groups; examples of Lie groups; some complex Lie groups; the three most commonly seen subgroups of Gl(n; ℂ): Sl(n; ℂ), U(n) and ... More


Preliminary Results and Discussion

Shaun M. Fallat and Charles R. Johnson

in Totally Nonnegative Matrices

Published in print:
2011
Published Online:
October 2017
ISBN:
9780691121574
eISBN:
9781400839018
Item type:
chapter
Publisher:
Princeton University Press
DOI:
10.23943/princeton/9780691121574.003.0002
Subject:
Mathematics, Applied Mathematics

This chapter lays out a number of basic and fundamental properties of TN matrices along with a compilation of facts and results from core matrix theory that are useful for further development of this ... More


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