Lecture 9
continued proof of Cayley-Hamilton. Applications: (i) Cayley-Hamilton to reduce the order of a polynomial in A, with example.
Lecture 10
(ii) Cayley-Hamilton to determine analytic functions of a matrix, with example. (iii) Computing exp(At), with example.
Lecture 11 and 12
Discussion of method for (iii) if 1 or more eigenvalues are repeated. Hermition Matrices: definition and example. Discussion of notation and recap of complex conjugation. Properties of hermitian matrices eg all real diagonal elements. Examples including Pauli matrices. Eigenvalues of a Hermitian matrix. Definition of anti-hermitian and unitary matrices.