Lecture 25
Diagonalisation - proof by construction. Link with linear-independence of eigenvectors.
Lecture 26
Applications: finding solutions to coupled differential equations. Remark on complex eigenvalues. Proof that symmetric matrices have only real eigenvalues.
Lecture 27 and 28
Orthogonal matrices and diagonalisation. Definition of orthogonal matrices and examples including rotation matrices. Recap of orthonormal bases and proof relation orthogonality and orthonormal basis of row vectors. Diagonalisation of orthogonal matrices (including Gram-Schmidt method). Examples.