Linear Algebra Diagonalization
A first course in linear algebra is an introductory textbook designed for university sophomores and juniors.
Linear algebra diagonalization. Most of this article focuses on real and complex matrices that is matrices whose elements are real numbers or complex numbers. Early in chapter vs we prefaced the definition of a vector space with the comment that it was one of the two most important definitions in the entire course here comes the other. An exception is the computation of norms which also extends to scalars and vectors. Further information on these functions can be found in standard mathematical texts by such authors as golub and van loan or meyer. V v is called diagonalizable if there exists an ordered basis of v with respect to which t is represented by a diagonal matrix.
In linear algebra a square matrix a is called diagonalizable if it is similar to a diagonal matrix ie if there exists an invertible matrix p such that p 1 ap is a diagonal matrix. The present textbook is intended for a one term course at the junior or senior level. A matrix is a rectangular array of numbers or other mathematical objects for which operations such as addition and multiplication are defined. This course covers matrix theory and linear algebra emphasizing topics useful in other disciplines such as physics economics and social sciences natural sciences and engineering. Most commonly a matrix over a field f is a rectangular array of scalars each of which is a member of f.
The book begins with systems of linear equations then covers matrix algebra before taking up finite dimensional vector spaces in full generality. It begins with an exposition of the basic theory of finite dimensional vector spaces and proceeds to explain the structure theorems for linear maps including eigenvectors and eigenvalues quadratic and hermitian forms diagonalization of symmetric hermitian and unitary linear maps and matrices. This note covers the following topics. It parallels the combination of theory and applications in professor strangs textbook introduction to linear algebra. Linear algebra matrix algebra homogeneous systems and vector subspaces basic notions determinants and eigenvalues diagonalization the exponential of a matrix applicationsreal symmetric matrices classification of conics and quadrics conics and the method of lagrange multipliers normal modes.
This tutorial reviews the functions that mathematica provides for carrying out matrix computations. This section contains a complete set of video lectures on linear algebra along with transcripts and related resource files. The operations described in this tutorial are unique to matrices. Students are expected to be comfortable with numerical linear algebra and multivariate calculus and to have programming experience preferably in matlab.