Advanced Linear Algebra, Lecture 4.1: Eigenvalues and eigenvectors

Advanced Linear Algebra, Lecture 4.1: Eigenvalues and eigenvectors Throughout this lecture and the others in this section, we will assume that X is a vector space over an algebraically closed field K, like the complex numbers. This ensures that every polynomial in K[x] has a root in K. An eigenvector of a linear map A:X→X is any nonzero v such that Av=λv, for some scalar λ called an eigenvalue. Equivalently, v is in the null space of A-λI, and so one characterization of an eigenvalue is any scalar for which det(A-λI)=0. This is how we actually find eigenvalues, and then we find the eigenvector by solving (A-λI)v=0. We show why every linear map has an eigenvector, and why eigenvectors corresponding to distinct eigenvalues are linearly independent. If X has a basis of eigenvectors of A, then we say that A is diagonalizable, because the matrix with respect to this basis is diagonal. Course webpage: http://www.math.clemson.edu/~ macaule/math8530-online.html

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