
Advanced Linear Algebra And Multilinear Algebra

There was once a professor who taught at this school who was…really something else. I mean this guy would show up to his class and meetings completely wasted, it wasn’t a good thing. However, he had an amazing talent for multiplying matrices in his head. I’m talking like 6x6 matrices. I got to wondering ‘wow, how could somebody do something like that,’ and then I remembered that when you are intoxicated, your left eye can move right to left, and your right eye can move up and down.— Saburo Matsumoto, linear algebra professor
 Advanced Linear Algebra Donald Estep Colorado State University Course Materials (PG13)
 This is another excellent,creative and comprehensive set of handwritten lecture notes by Estep these either for an honors course or a second course in linear algebra. These notes, unlike many at this level, don't require any specific background in abstract algebra, although the equivalent of a proofbased
 undergraduate course in linear algebra is a musthave. Student looking at these notes may be a bit surprised by the content.Students looking for an abstract presentation of vector spaces as free R modules are going to have to look elsewhere there are plenty of good choices in the undergraduate and graduate
 abstract algebra sections for sources like this. Algebraic structuresother then vector spaces and their scalar fields, of courseplay nearly no role in this presentation since their focus is on the role of linear operators and their spaces in analysis, particularly function and normed spaces, matrix analysis and
 differential equations. Appropriately, the main textbooks for the course were Peter Lax's wonderful book, which has become the gold standard for such "vector spaces for analysis" courses and the standard text on hard analysis over matrices, Matrix Analysis, 2nd edition by Roger A. Horn and Charles R. Johnson .
 It's clear it's important to offer such courses. These aspects of linear algebra are usually not covered in the typical advanced linear algebra or graduate level algebra courses. But they are just as important as the algebraic theory of vector spaces, particularly for graduate students looking to take serious courses in functional analysis,operator theory or advanced differential equations. Also,matrix analysis over abstract vector spaces is especially important in applied mathematics. In addition to the 2 main textbooks, there are over a dozen additional references the notes are distilled by Estep from all of these sources.The resulting notes form a textbook in and of themselves they are very clear and readable, with many examples and guiding comments from the author in his usual lucid and lively style. These notes will assist students greatly in navigating the enormous and important web that is the foundation of applied linear algebra. A large number of applications to both geometry and analysis are present. These notes will make a fine supplement for graduate studentsnot only as a supplement for a graduate algebra course showing the "other side" the theory of vector spaces, but particularly those who are planning to specialize
 in either applied algebra or analysis. Highly recommended for all students of linear algebra regardless of level. Contents: 1. Review and Fundamental Concepts 2. Duality 3. Linear transformations 4.
 Matrices 5. Determinant and Trace 6. Spectral Theory 7. Euclidean Spaces 8. Normed Linear Spaces ) 9. Unitary Equivalence and Normal Matrices 10. Jordan Canonical Form 11. PLU Decomposition 12.
 Singular Value Decomposition 13. Spectral Theory for Self Adjoint Maps
 Multilinear Algebra 1 TinYau Tam Auburn University 2011 (PG13) I remember Googling "multilinear algebra lecture notes" in the seemingly interminable research phase of preparing this website and remember being resigned to not expect to find much. Both the algebracists and the differential geometers I spoke to were of the consensus that "almost no one teaches courses in multilinear algebra anymore, that material's covered in either the graduate algebra or graduate differential geometry courses if at all." Which is why the author of these notes got me to hello on this. Not only did he apparently give such a course, not only did he write his own notes as the primary textbook for the course, but they're very substantial, readable and touch on virtually all areas of pure and applied mathematics (except differential geometry, which is understandable) where tensor products and multilinear maps play a significant role. Contents: Review of Linear Algebra Basics of finite group representation theory Multilinear maps and tensor spaces Symmetry class of tensors Generalized matrix functions Applications and current
 research I'm not sure what the prerequisites for the course where the website isn't clear. I'd imagine at the very least, students would need either an intensive one year course in linear algebra or an equally intensive one year course in abstract algebra at the honors level. I don't quite think the usual one semester "baby" courses in linear and abstract algebra are going to be sufficient since the review
 section pulls in topics that you don't normally see in those coursessuch as dual spaces and direct sums of vector spaces. which are very reasonable for an indepth presentation of this material. I think the ideal students for this course would be either late first year graduate students or very strong undergraduates applying to top graduate programs in mathematics. They're about the same level as Gulliemin's
 notes on differential forms, but these notes are about the general subject of multilinear algebra and forms play only a very small role here. Tam's notes are extremely broad in scope and cover a lot of ground. Universal properties of tensor spaces are discussed at length in it's own chapter.There are many examples and detailed computations with tensor products, direct sums and group representations. Indeed, the chapter on group representations from the standpoint of tensor algebra is the high point of the notes and it could easily function as the material on representation theory of finite groups in a
 year long graduate algebra course But there is so much more here, including many applications to algebra, combinatorics and much more. There are also a surprisingly large number of exercises, of diverse difficulty. Tam's notes are a real find, one of the last of a dying breed of course notes on this truly important area of algebra. I don't know how much longer they'll be available for download. So I recommend immediate download of all the materials at Tam's page immediatelyyou'll be glad to have them on hand if you're a graduate student in mathematics, trust me. Very highly recommended.
 A Second Semester of Linear Algebra S. E. Payne University of Colorado at Denver 2009 (PG13) A broad,concise and readable set of notes for an advanced linear algebra course for students with backgrounds in both basic linear algebra and undergraduate abstract algebra.Payne focuses on careful definitions and proofs of theorems, with very few examples.There aren't even many examples in the exercises.This is a matter of taste, as I've said elsewhere at this site, but that's a big red flag for me
 despite finding the notes quite readable and lucid. An unusual touch is that Payne gives the vector space of nvariable polynomials over an arbitrary field F a major role and motivates many of the central
 concepts through this particular space.Lots of good exercises, none too difficult. It also contains more advanced material not usually covered in courses at this level, such as good introductions to the
 general study of matrix functions and the bare bones of infinite dimensional vector spaces and how they differ from their finite dimensional counterparts.I think the latter is particularly useful for students to see in a course like this because in a functional analysis course, these basic differences often get overlooked or treated too causally to be fully understood. Sadly, many of the standard topics that used to be covered in linear algebra are also coveredsuch as diagonalization and subspacesshowing how diluted such courses have generally become at the undergraduate level. Still,overall it's a solid course on this material, worth looking at for a second course with the appropriate background. Recommended.Contents: 1 Preliminaries 2 Vector Spaces 3 Finite Dimensional Vector Spaces 4 Linear Transformations 5 Polynomials 6 Determinants 7 Operators and Invariant Subspaces 8 Inner Product Spaces 9 Operators on Inner Product Spaces 10 Decomposition WRT a Linear Operator 11 Matrix Functions 12 Infinite Dimensional Vector Spaces
 A Comprehensive Introduction to Linear And Multilinear Algebra Joel G. Broida and S. Gill Williamson/Multilinear Algebra Seminar Marvin Marcus USCD This staggeringly comprehensive textbook on linear algebra was first published in 1986 and has been reposted as 3 large PDF
 files at the authors' website. It's really amazing that this book has been out of print as long as it has and the authors have done a great service to students making it available again online. The book was originally developed from diverse sets of lecture notes for various graduate courses at USCD over many
 years. The twofold purpose of the book was to a) gather into one source all the linear algebra that graduate students in mathematics, physics, engineering and computer science need to master to be able to do research in any of those areas and b)to have all this material available in a single unified presentation to students of various backgrounds not only as a course text but for career reference as needed. The book succeeds admirably on both counts.
 Linear Algebra William Kahan University of California Berkeley Fall semester 2000, and for Math. 110 Spring Semester 2002 (PG13)These intensive notes were complied over nearly 2 decades of
 teaching both the regular and honors linear algebra course at Berkeley by a distinguished expert in numerical linear algebra. They are similar to Froese and Wetton’s notes above in that they strongly
 emphasize the connection between basic linear algebra and classical geometry, but they are much more abstract,sophisticated and take a much more modern bent. The big flaw in these notes is that the
 subject matter is very disorganized despite being dated on each PDF file: the subject matter seems to range all over the place without much of a plan or logic. I think this is a result of being compliled
 over many years for several different courses at different levels and emphasis. I wish the author had taken some time to arrange and partition the notes by subject matter and difficulty level. Still,
 the sheer range and depth of the notes is very impressive. They include many topics one does not see in general linear algebra courses at either the undergraduate or graduate level, such as Chio’s algorithm for solving integral coefficient linear systems, domination in diagonal matrices, Jacobi's Formula for the derivative of a determinant function, Jensen’s inequality, a lengthy and lucid section on the Jordan form and its derivation, Hilbert forms and much more. But by far the best thing about these notes are the wonderful, detailed and highly personal historical and referential sections on each topic that accompany
 most of the notes and which are in some ways more informative than the notes themselves. Some memorable excerpts: “Likely”, “usually”, “most”, “almost all”, “nearly” … are weasel words that offend many a pure mathematician who prefers “always”, “every”, “all” … and takes “exactly” for granted. Applied mathematicians encounter situations often where the weasel words are the best that can be expected. In fact, the weasel words above are provably the best that can be expected in their contexts, but that is a story for another day.The noun “functional” arose first from the adjective in Functional Analysis, which was at first concerned with operators that map functions to scalars; an instance is the definite integral h(ƒ) = òo 1 ƒ(K) dK regarded as an operator upon functions ƒ(K) . In time, each function ƒ(K) became identified with a point and then a vector f in a space of functions. Then it became possible to write h(ƒ) = hTf where the linear functional hT stood for the integral operator òo 1 …(K) dK divorced from the function ƒ upon which it acted. Note how the functional hT is not the transpose of a
 vector h ; there is no vector h . Pedants who wished to stress this nonrelationship used to choose, say, vT
 for a functional and v for an unrelated function, like a playwright choosing names “Edwin” and
 “Edwina” for unrelated characters. That’s a confusing choice I intend to avoid.There is no way to
 exhibit a Hamel basis because there is no way to decide for every set of Reals whether it is linearly independent. Like Spiritualism and Physics, Mathematics has its invisible presences.In 1982 Bill Gates Jr. predicted that almost no IBM PC’s socket for Intel’s 8087 Numeric Coprocessor would ever be filled,
 so no good reason existed to change Microsoft’s compilers to support all three of the 8087’s floating point formats. He was quite wrong on both counts, but Microsoft’s compilers still eschew its widest format. In the mid1980s the ANSI X3J11 committee responsible for standardizing C acquiesced to demands from CDC and Cray Research to let C compiler writers choose FORTRANnish expression evaluation instead of
 Kernighan Ritchie’s. That undid their serendipity and accelerated the migration of scientific and engineering computations from floats to doubles on all computers but now defunct CDC’s and Cray Research’s. Early in the 1990s, just as programmers were beginning to appreciate SANE and praise it, John Sculley tried to put Apple into bed with IBM and switched Macintoshes to IBM’s Power PC microprocessor although it could not support SANE on Power Macs. That liaison’s Taligent Inc. lived only briefly; and recently Apple switched Macintoshes to Intel’s microprocessors. These could support SANE but Macs don’t yet; Apple’s efforts focus now upon a far bigger market. In the mid 1990s James
 Gosling and Bill Joy at Sun Microsystems invented the programming language Java partly to cure C of pointer abuse but mostly to break Microsoft’s stranglehold upon the computing industry. They pointedly avoided consultation with Sun’s numerical experts when they adopted FORTRANnish expression evaluation instead of KernighanRitchie’s, and banned extraprecise arithmetic and any requirement of IEEE 754 they didn’t like. Java’s floatingpoint is dangerous; see “How Java’s FloatingPoint Hurts
 Everyone Everywhere”I think except for the most elementary notes in the set, most of these are simply going to be too difficult for most undergraduates who haven’t had at leasta solid course in linear algebra. But for students that have that background, they’re a gold mine of modern linear
 analysis that would benefit all by working through. Recommended for advanced students.
 Exercises and Problems in Linear Algebra John M. Erdman Portland State University Version September 18, 2011 (PG13/R) This is one of several excellent advanced online textbooks inprogress that Erdman’s made freely available at his website. These are his notes for an advanced course on linear and multilinear
 algebra for first year graduate students with some undergraduate background in abstract and linear algebra. There aren’t many advanced standalone treatments of linear algebra, both in print and
 online (although in fairness, that number has begun to significantly increase in recent years). Most advanced treatments of linear algebra are either matrix analysis courses or are part of the honors
 undergraduate/ graduate course in abstract algebra. But there are even fewer on multilinear and tensor algebra as an offshoot of the study of vector spaces. Most of the ones that do exist such as Blyth and Northcott, which are both excellentare over 30 years old. So a modern, unified treatment of both subjects is extremely desirable. Erdman is a devotee of the mathematicial philosophy that students learn
 mathematics by doing mathematicshence, the book is quite terse and a lot of details are left to the reader.In fact, it’s essentially a problem course with a lot of hints and definitions. This isn’t necessarily a bad thing when you’re dealing with advanced students. A lot of mathematicians think this is the way to
 teach mathematics. I’ll discuss this in more detail on my blog in the future, but here I’ll just say too many mathematicians and teachers use this as an excuse to make the students figure the coursework out on their own while they go off and do more important things than teaching. Erdman doesn’t do that here.
 Firstly, he gives an enormous number of examples and most of them are quite simple. This allows
 him to state them in minimal detail without any harm done and simultaneously giving the students lots to chew on by their sheer number. He also has nearly the same number of exercises both implicit through minor results he leaves for the student to prove and explicitly labeled as such completely interwoven with the textbook material proper. Such a text is clearly intended to be read with pencil and paper in hand and as such, it can be a joy for advanced students to work through provided the “exercises” are not unreasonably difficult. From the reading I did, most don’t appear very difficult for the intended audience level. Thirdly, the notes are extremely modern and inventive. For example introduces the language of category theory in chapter 3 and uses it throughout. Category theory and linear algebra are as natural (pun intended) a pairing as fried chicken and cornbread and I’m surprised more authors don’t use it outside of a graduate course in algebra. Spectral theory is discussed in greater detail then you normally see in either advanced linear algebra texts or abstract algebra texts. Multilinear algebra is developed in great detail, both it’s algebraic and geometric aspects, including tensor products differential forms and an introduction to Clifford algebras. In short, this is a beautifully written,very versatile and challenging text on this central subject for advanced students which professors can choose to use at varying levels by filling in as much or as little detail as they like. Highly recommended.
Linear Algebra II Diagonalization Theory Issac Freed Boston University 2012 (PG13)This is chapter 6 of Freed's extensive linear algebra notes, the only chapter that still remains accessible online at this writing. The author has taken the other chapters down whether to begin the process of revising them for future  coursework/publication or because of strictly ideological reasons"no free lunch from my notes,pay the f*****g tuition if you want to see them!"I have no idea. Hopefully it's the former. In any event, the part of the notes that survives online is on finite dimensional spectral theory (i.e. the theory of the eigenvalues of characteristic polynomials and thier corresponding eigenvectors) and it's very good indeed. It's really important to master this part of basic linear algebra because the generalization of these methods is of such central importance in functional analysis and it's applications to the physical sciences. So an elementary set of notes that focuses entirely on it is a welcome resource. The author is serious about the subject and ties it without hesitation to basic analysis and geometry. For example, he motivates the
 subject by using the extreme value theorem on a compact surface in R^{3} to set a minimum bound on the "degree of independence" the vectors in a basis. While this is interesting and creative, I wonder how many students, even good ones, will be able to follow the reasoning here.Towards the end, he discusses the elements of infinite dimensional spectral theory. So these are serious notes for strong mathematics
 students.But overall, I found Freed's notes very dense, disorganized and lacking in examples. I like the notes in some places and the depth of the author's presentation is praiseworthy. But I doubt these notes could be used without their predecessor notes to set both background and notation. You can check them out, but I'd rather use Lehner or Treil.
 Graduate Linear Algebra Jim Brown Clemson University Fall 2012 (PG13) These are one of a large collection of handwritten lecture notes by Jim Brown ( an associate professor at Clemson University, not the old football legendjust wanted to clear that up) complied from his personal collection of classroom notes he either took down as a graduate student or in courses he either taught or cotaught at various
 universities when he was a PHD student and postdoc. This particular set is from a graduate course in linear algebra. It’s tricky to post scans of your handwritten notes. There are 2 potential problems; a)
 the scans themselves may be too dark or fuzzy to be clear or b) your handwriting may leave much to be desired. The latter is generally the much more serious issue the most complete and insightful lecture notes does no one any damn good if your handwriting looks like they were taken down by a drunken, half blind gorilla. And of course, typing up your lectures afterwards is incredibly time consuming. (Live TeXing your notes is becoming a very good and common solutionI’m itching to try it myself when my TeX skills are up to speed.) But if you plan on doing it the old fashioned way and scanning them in before posting them, you better make sure your handwriting is legible. Brown’s handwriting isn’t bad and if you
 make a little effort, you can generally read them just fine. But there are spots where the notes are either too lightly or too darkly scanned. As far as the content goes, the notes cover all the standard material
 of an undergraduate linear algebra course at a more advanced level, assuming some knowledge of abstract algebra. The notes prove most results in full detail with many examples the section on spectral theory and the Jordan form are particularly nice and useful. Canonical forms are used throughout the final sections in many applications that are interesting. There are sections where his notation is confusing and there are some outright errors or illegible parts here and there. But if you have the patience, they’re very diligently and insightfully written and will be an asset to graduate students. Recommended.
 ADVANCED NOTES ON LINEAR ALGEBRA LIVIU I. NICOLAESCU UNIVERSITY OF NOTRE DAME
2012 (PG13) These are Nicolaescu's notes for an advanced linear algebra course at the advanced  undergraduate or graduate level, for students who already have an introductory course behind them. CONTENTS 1 Multilinear forms and determinants 2 Spectral decomposition of linear operators 3
 Euclidean spaces 4 Spectral theory of normal operators 5 Applications 6. Elements of linear topology References Like all of Nicolaescu's notes, these are extremely readable without being too pedantic and contain many examples and exercises. An original touch here is that he makes a real effort to connect the
 sophisticated concepts of advanced linear space theory to the more basic and conventional elements. For example, he develops the theory of the determinant of an n by n matrix as a skewsymmetric multilinear mapping on F^{n} where F is the chosen field of a vector space V. The development of multilinear maps is quite clear and even better, relatively brief. Many topics that are usually covered in more advanced algebra or functional analysis courses are developed here in a very clear and organized way. The development of spectral theory of linear operators in finite dimensional vector spaces in particular is outstanding and will make very good background reading before taking a graduate analysis course. For
 students looking for the next step after a basic semester of linear algebra, they could do a lot worse then these notes. Highly recommended as a second course.