Linear Algebra Diagonalization
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.
Linear algebra diagonalization. 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. Further information on these functions can be found in standard mathematical texts by such authors as golub and van loan or meyer. Any capsule summary of linear algebra would have to describe the subject as the interplay of linear transformations and vector spaces. A matrix is a rectangular array of numbers or other mathematical objects for which operations such as addition and multiplication are defined. Students are expected to be comfortable with numerical linear algebra and multivariate calculus and to have programming experience preferably in matlab.
V v is called diagonalizable if there exists an ordered basis of v with respect to which t is represented by a diagonal matrix. 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. A first course in linear algebra is an introductory textbook designed for university sophomores and juniors. Most of this article focuses on real and complex matrices that is matrices whose elements are real numbers or complex numbers. This tutorial reviews the functions that mathematica provides for carrying out matrix computations.
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. The operations described in this tutorial are unique to matrices. This note covers the following topics. It parallels the combination of theory and applications in professor strangs textbook introduction to linear algebra. If v is a finite dimensional vector space then a linear map t.
The present textbook is intended for a one term course at the junior or senior level. 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. The book begins with systems of linear equations then covers matrix algebra before taking up finite dimensional vector spaces in full generality. Typically such a student will have taken calculus but this is not a prerequisite.