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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

## 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

## 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

## 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

## 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

## 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

## 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

## Quarter Turn Compositions and PETs

*Richard Evan Schwartz*

### in The Plaid Model: (AMS-198)

- Published in print:
- 2019
- Published Online:
- September 2019
- ISBN:
- 9780691181387
- eISBN:
- 9780691188997
- Item type:
- chapter

- Publisher:
- Princeton University Press
- DOI:
- 10.23943/princeton/9780691181387.003.0016
- Subject:
- Mathematics, Educational Mathematics

This chapter proves a general compactification theorem for quarter turn compositions. It is organized as follows. Section 15.2 proves a well-known result from linear algebra which will help with the ... More

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