Introduction to Linear Algebra with Applications

These notes were written as a part of a graduate level course on transform theory offered at King's College London during 2002 and 2003. The material is heavily indebt to the excellent textbook by Gilbert Strang [1], which the reader is referred to for a more complete description of the material; for a more in-depth coverage, the reader is referred to [2–6].

The American Mathematical Monthly, 1969

The purpose of this first set of lecture notes is to summarize background material on linear algebra and analysis that is used throughout the course. Proofs of most of the stated facts may be found in the references listed at the end of the notes. It is my intention that this first set of notes will mostly serve as a useful reference later in the course.

an asterisk can be omitted without loss of continuity but may be required for later optional sections. See each indicated section for dependency information.

Linear Algebra and its Applications, 1992

This two volume set of texts is designed to provide material for a course of up to one full year at the undergraduate level. It requires as a prerequisite some knowledge of calculus and complex numbers, but little mathematical

2007

S Subadditive and Superadditive Inequalities T. Ando Faculty of Economics Hokusei Gakuen University Sapporo 004-8631 Japan [email protected] Key identifying words: subadditive and superadditve inequalities, operator-monotone function. It is known that if f(t) is a non-negative, operator-monotone (that is, matrix-monotone of all orders) function on [0;1) then the inequality jjjf(A+B) f(A)jjj jjjf(B)jjj holds for every pair of positive semi-de nite matrices A;B and every unitarily invariant norm jjj jjj. In this talk we discuss inequalities of the form jjjf(A+B)jjj jjjf(A) + f(B)jjj: Diagonals of Matrices Rajendra Bhatia Indian Statistical Institute New Delhi 110 016 India [email protected] Key identifying words: diagonal, norm, Fourier series, inequalities The diagonal of a matrix can be expressed as an average over unitary conjugates of the matrix.We will display such expressions for the main diagonal and other diagonals parallel to it.This leads to some interesting sharp bounds...

Linear Algebra and its Applications, 1983

This unusual little book, which is intended to serve both as a text and as a reference, is a survey of a broad portion of the theory of linear spaces, with particular attention to the solution of linear equations. The book is in four chapters, entitled "Nontopological linear space theory," "Finite systems of linear algebraic equations and their generaliTations," "Topological linear spaces: some comparisons," and "Current research problems." It differs from other brief books on these subjects in two principle ways. First, the author takes a very broad view, putting into 168 pages the beginnings or the essentials of a wide variety of topics, including the general theory of linear spaces, systems of linear equations, integral operators and integral equations, compact operators, the Fredholm theorems and the theories of Banach spaces, of Hflbert spaces, and of topological vector spaces. Second, the book is built on a very systematic and explicit comparison of the finite and infinite dimensional theories. In much of the book, this comparison is undertaken on a theorem by theorem basis and most results are presented in two different versions, with comments on their differences. Where no reasonable generalization to the infinite dimensional case exists, the book generally provides relevant examples in considerable detail. The author's goal seems to have been that of illuminating a large area of mathematics by means of this systematic comparison. Generalizing a theorem is usually an excellent way of coming to a firm understanding of it, so Professor Jfi_rvinen's approach has much to recommend it. Moreover, this book contains an accessible compilation of a large number of results, from different but related areas, which otherwise would not be easy to find in a single source. Unfortunately, no treatment of so much in so short a space can possibly be complete. Professor Jfi_rvinen is thus obliged to replace most of the proofs by references, and to limit himself largely to supplying a context and to describing the flow of the results in his subjects. This approach poses the danger that the exposition can easily turn into a loosely connected list of definitions and results. On the other hand, it makes for a very clear view of broad outlines, and it places on the reader the burden of supplying or

The principal change from the first edition is the addition of a new chapter on linear programming. While linear programming is one of the most widely used and successful applications of linear algebra, it rarely appears in a text such as this. In the new Chapter Ten the theoretical basis of the simplex algorithm is carefully explained and its geometrical interpretation is stressed.

Lecture Notes in Mathematics, 1979

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We present new comparison theorems for the spectral radii of matrices arising from splittings of different matrices under nonnegativity assumptions. Our focus is on establishing strict inequalities of the spectral radii without imposing strict inequalities of the matrices, but we also obtain new results for nonstrict inequalities of the spectral radii. We emphasize two different approaches, one combinatorial and the other analytic and discuss their merits in the light of the results obtained. We try to get fairly general results and indicate by counter-examples that some of our hypotheses cannot be relaxed in certain directions.