SeriesSeriesSeries ISSN: ISSN: ISSN: 1938-1743 1938-1743 1938-1743 TOBIAS TOBIAS TOBIAS SSSYNTHESISYNTHESISYNTHESIS L LLECTURESECTURESECTURES ON ONON MMM MorganMorganMorgan & & & Claypool ClaypoolClaypool Publishers PublishersPublishers MMMATHEMATICSATHEMATICSATHEMATICS AND AND AND S S STATISTICSTATISTICSTATISTICS &&&CCC SeriesSeriesSeries Editor: Editor: Editor: Steven Steven Steven G. G. G. Krantz, Krantz, Krantz, Washington Washington Washington University, University, University, St. St. St. Louis Louis Louis MatricesMatricesMatrices in inin Engineering EngineeringEngineering Problems ProblemsProblems MatricesMatricesMatrices in inin MarvinMarvinMarvin J. J. TobiasJ. Tobias Tobias MATRICES IN ENGINEERING PROBLEMS MATRICES IN MATRICES ENGINEERING IN PROBLEMS ENGINEERING PROBLEMS ThisThisThis book book book is is intendedis intended intended as as anas an anundergraduate undergraduate undergraduate text text text introducing introducing introducing matrix matrix matrix methods methods methods as as theyas they they relate relate relate to to engi-neeringto engi-neering engi-neering EngineeringEngineeringEngineering Problems ProblemsProblems problems.problems.problems. It It beginsIt begins begins with with with the the the fundamentals fundamentals fundamentals of of mathematicsof mathematics mathematics of of matricesof matrices matrices and and and determinants. determinants. determinants. Matrix Matrix Matrix inversion inversion inversion is isdiscussed,is discussed, discussed, with with with an an anintroduction introduction introduction of of ofthe the the well well well known known known reduction reduction reduction methods. methods. methods. Equation Equation Equation sets sets sets are are are viewed viewed viewed as as as vectorvectorvector transformations, transformations, transformations, and and and the the the conditions conditions conditions of of oftheir their their solvability solvability solvability are are are explored. explored. explored. OrthogonalOrthogonalOrthogonal matrices matrices matrices are are are introduced introduced introduced with with with examples examples examples showing showing showing application application application to to tomany many many problems problems problems requiring requiring requiring threethreethree dimensional dimensional dimensional thinking. thinking. thinking. The The The angular angular angular velocity velocity velocity matrix matrix matrix is is shownis shown shown to to toemerge emerge emerge from from from the the the differentiation differentiation differentiation ofof ofthe the the 3-D 3-D 3-D orthogonal orthogonal orthogonal matrix, matrix, matrix, leading leading leading to to tothe the the discussion discussion discussion of of ofparticle particle particle and and and rigid rigid rigid body body body dynamics. dynamics. dynamics. TheTheThe book book book continues continues continues with with with the the the eigenvalue eigenvalue eigenvalue problem problem problem and and and its its itsapplication application application to to tomulti-variable multi-variable multi-variable vibrations. vibrations. vibrations. BecauseBecauseBecause the the the eigenvalue eigenvalue eigenvalue problem problem problem requires requires requires some some some oper oper operationsationsations with with with polynomials, polynomials, polynomials, a aseparate a separate separate discussion discussion discussion of of of thesethesethese is is givenis given given in in inan an anappendix. appendix. appendix. The The The example example example of of ofthe the the vibrating vibrating vibrating string string string is is givenis given given with with with a acomparison a comparison comparison of of ofthe the the matrixmatrixmatrix analysis analysis analysis to to tothe the the continuous continuous continuous solution. solution. solution. MarvinMarvinMarvin J. J.J. Tobias TobiasTobias AboutAboutAbout SYNTHESIs SYNTHESIs SYNTHESIs Morgan MorganMorgan ThisThisThis volume volume volume is is ais aprinted a printed printed version version version of of ofa awork a work work that that that appears appears appears in in inthe the the Synthesis Synthesis Synthesis DigitalDigitalDigital Library Library Library of of Engineeringof Engineering Engineering and and and Computer Computer Computer Science. Science. Science. Synthesis Synthesis Synthesis Lectures Lectures Lectures provideprovideprovide concise, concise, concise, original original original presentations presentations presentations of of importantof important important research research research and and and development development development & & & topics,topics,topics, published published published quickly, quickly, quickly, in in digitalin digital digital and and and print print print formats. formats. formats. For For For more more more information information information Claypool Claypool Claypool visitvisitvisit www.morganclaypool.com www.morganclaypool.com www.morganclaypool.com ISBN:ISBN:ISBN: 978-1-60845-658-1 978-1-60845-658-1978-1-60845-658-1 909900000000 SSSYNTHESISYNTHESISYNTHESIS L LLECTURESECTURESECTURES ON ONON MorganMorganMorgan& & & Claypool ClaypoolClaypool Publishers PublishersPublishers ATHEMATICSATHEMATICSATHEMATICS AND AND AND TATISTICSTATISTICSTATISTICS www.morganclaypool.comwww.morganclaypool.comwww.morganclaypool.com MMM SSS 99798771886110668008485446556685518811 StevenStevenSteven G. G. G. Krantz, Krantz, Krantz, Series Series Series Editor Editor Editor Matrices in Engineering Problems Synthesis Lectures on Mathematics and Statistics Editor Steven G. Krantz, Washington University, St. Louis Matrices in Engineering Problems Marvin J. Tobias 2011 The Integral: A Crux for Analysis Steven G. Krantz 2011 Statistics is Easy! Second Edition Dennis Shasha and Manda Wilson 2010 Lectures on Financial Mathematics: Discrete Asset Pricing Greg Anderson and Alec N. Kercheval 2010 Jordan Canonical Form: Theory and Practice Steven H. Weintraub 2009 The Geometry of Walker Manifolds Miguel Brozos-Vázquez, Eduardo García-Río, Peter Gilkey, Stana Nikcevic, and Rámon Vázquez-Lorenzo 2009 An Introduction to Multivariable Mathematics Leon Simon 2008 Jordan Canonical Form: Application to Differential Equations Steven H. Weintraub 2008 iii Statistics is Easy! Dennis Shasha and Manda Wilson 2008 A Gyrovector Space Approach to Hyperbolic Geometry Abraham Albert Ungar 2008 Copyright © 2011 by Morgan & Claypool All rights reserved. No part of this publication may be reproduced, stored in a retrieval system, or transmitted in any form or by any means—electronic, mechanical, photocopy, recording, or any other except for brief quotations in printed reviews, without the prior permission of the publisher. Matrices in Engineering Problems Marvin J. Tobias www.morganclaypool.com ISBN: 9781608456581 paperback ISBN: 9781608456598 ebook DOI 10.2200/S00352ED1V01Y201105MAS010 A Publication in the Morgan & Claypool Publishers series SYNTHESIS LECTURES ON MATHEMATICS AND STATISTICS Lecture #10 Series Editor: Steven G. Krantz, Washington University, St. Louis Series ISSN Synthesis Lectures on Mathematics and Statistics Print 1938-1743 Electronic 1938-1751 Matrices in Engineering Problems Marvin J. Tobias SYNTHESIS LECTURES ON MATHEMATICS AND STATISTICS #10 M &C Morgan& cLaypool publishers ABSTRACT This book is intended as an undergraduate text introducing matrix methods as they relate to engi- neering problems. It begins with the fundamentals of mathematics of matrices and determinants. Matrix inversion is discussed, with an introduction of the well known reduction methods. Equation sets are viewed as vector transformations, and the conditions of their solvability are explored. Orthogonal matrices are introduced with examples showing application to many problems requiring three dimensional thinking. The angular velocity matrix is shown to emerge from the differentiation of the 3-D orthogonal matrix, leading to the discussion of particle and rigid body dynamics. The book continues with the eigenvalue problem and its application to multi-variable vi- brations. Because the eigenvalue problem requires some operations with polynomials, a separate discussion of these is given in an appendix. The example of the vibrating string is given with a comparison of the matrix analysis to the continuous solution. KEYWORDS matrices , vector sets, determinants, determinant expansion, matrix inversion, Gauss reduction, LU decomposition, simultaneous equations, solvability, linear regression, orthogonal vectors & matrices, orthogonal transforms, coordinate rotation, Eulerian angles, angular velocity and momentum, dynamics, eigenvalues, eigenvalue analysis, characteristic polynomial, vibrating systems, non-conservative systems, Runge-Kutta integration vii Contents Preface .................................................................xiii 1 Matrix Fundamentals ......................................................1 1.1 Definition of A Matrix .................................................. 1 1.1.1 Notation ........................................................ 2 1.2 Elemetary Matrix Algebra ............................................... 3 1.2.1 Addition (Including Subtraction) ................................... 4 1.2.2 Multiplication by A Scalar ........................................ 4 1.2.3 Vector Multiplication ............................................. 4 1.2.4 Matrix Multiplication ............................................ 6 1.2.5 Transposition .................................................... 8 1.3 Basic Types of Matrices .................................................