Topics in Linear Algebra - The Functional Calculus  - 05 - Proof of main theorem
Channel: Arthur Parzygnat
Date: 06/10/2019

We prove our main theorem for the functional calculus. Recall, this says that for every diagonalizable matrix A and any function f on the set of eigenvalues of A, there exists a polynomial q such that f(A)=q(A). We also prove an interesting corollary that says f(A)B=Bf(A) whenever AB=BA.

Note: I don't stress this enough in the videos, but it is important to understand the quantifiers here: the polynomial q depends on both f and A. If we change our matrix A, we may need to find a new polynomial! 

This is part of a series of lectures on special topics in linear algebra https://www.youtube.com/playlist?list=PLSx1kJDjrLRSS4Aio9WfQlNnH6X6pC_8t. It is assumed the viewer has taken (or is well into) a course in linear algebra. Some topics may also require additional background. Topics covered include linear regression and data analysis, ordinary linear differential equations, differential operators, function spaces, category-theoretic aspects of linear algebra, support vector machines (machine learning), Hamming's error-correcting codes, stochastic maps and Markov chains, tensor products, finite-dimensional C*-algebras, algebraic probability theory, completely positive maps, aspects of quantum information theory, and more.

The next video on Special Topics in Linear Algebra changes to an independent topic, which is on affine combinations: https://youtu.be/fWRm9dISpNk

The previous video on Special Topics in Linear Algebra is on The determinant of a Vandermonde matrix: https://youtu.be/xjDN9kI_qlc

These videos were created during the 2019 Spring/Summer semester at the UConn CETL Lightboard Room.