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