Advanced Linear Algebra, Lecture 4.2: The Cayley-Hamilton theorem

Advanced Linear Algebra, Lecture 4.2: The Cayley-Hamilton theorem The characteristic polynomial of a linear map A:X→X is p(t)=det(λt-A), and its roots are the eigenvalues of A. In this lecture, we show how the determinant of A is the product of its eigenvalues, and the trace of any matrix of A is the sum of its eigenvalues. This establishes that the trace is a property of a linear map, rather than just of a matrix, which is how we originally defined it. We also show that for any polynomial q(t), the eigenvalues of q(A) are all of the form q(λ), where λ is an eigenvalue of A. As a corollary, if p(t) is the characteristic polynomial of A, then all of the eigenvalues of p(A) are zero. In fact, it turns out that a stronger statement is true: p(A) is the zero map. This is called the Cayley-Hamilton theorem, and we prove it in a basis-free manner, rather than in terms of matrices, as is often standard. Course webpage: http://www.math.clemson.edu/~ macaule/math8530-online.html

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