Fundamentals of Matrix Analysis with Applications

This book provides comprehensive coverage of matrix theory from ageometric and physical perspective, and the authors address thefunctionality of matrices and their ability to illustrate and aidin many practical applications. Readers are introduced toinverses and eigenvalues through physical examples such asrotations, reflections, and projections, and only then arecomputational details described and explored. MATLAB isutilized to aid in reader comprehension, and the authors arecareful to address the issue of rank fragility so readers are notflummoxed when MATLAB displays conflict with manualcalculations. Matrix foundations are exphasized via projectsinvolving LU factorizations and the matrix aspects of finitedifference modeling and Kirchhoff's circuit laws. Vectorspace concepts and the many facets of orthogonality are thendiscussed, and in an effort maintain a computational perpective,attention is directed to the numerical issues of error controlthrough norm preservation. Projects include rotationalkinematics, Householder implementation of QR factorizations, andthe infinite dimensional matrices arising in Haar waveletformulations. The statistical unlikeliness of singular squarematrices, multiple eignevalues, and defective matrices are thenemphasized for random matrices, and the basic workings of the QRalgorithm (and the role of luck in its implementation as well as inthe occurrence of defective matrices) and the random-shiftamelioration of its failures are explored. The book concludeswith a chapter on the role of matrices in the solution of linearsystems of diffential equations (DEs) with constant coefficientsvia the matrix exponential. Insight into the ssues related toits computation are also provided.