Linear algebra via exterior products
by Sergei Winitzki
Publisher: Ludwig-Maximilians University 2009
Number of pages: 82
A pedagogical introduction to the coordinate-free approach in basic finite-dimensional linear algebra. The reader should be already exposed to the elementary array-based formalism of vector and matrix calculations. In this book, the author makes extensive use of the exterior product of vectors. He shows how the standard properties of determinants, the Liouville formula, the Hamilton-Cayley theorem, and Pfaffians, as well as some results concerning eigenspace projectors can be derived without cumbersome matrix calculations.
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by Zico Kolter - Stanford University
From the tabble of contents: Basic Concepts and Notation; Matrix Multiplication; Operations and Properties; Matrix Calculus (Gradients and Hessians of Quadratic and Linear Functions, Least Squares, Eigenvalues as Optimization, etc.).
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The authors consider the two matrix model with an even quartic potential and an even polynomial potential. The main result is the formulation of a vector equilibrium problem for the limiting mean density for the eigenvalues of one of the matrices.
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Special Set Linear Algebras introduced by the authors in this free book is an extension of Set Linear Algebras, which are the most generalized form of linear algebras. These structures can be applied to multi-expert models.
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In this book, the authors introduce the notion of Super linear algebra and super vector spaces using the definition of super matrices defined by Horst (1963). This book expects the readers to be well-versed in linear algebra.