Advanced Engineering Mathematics, Lecture 4.2: Symmetric and Hermitian matrices

Advanced Engineering Mathematics, Lecture 4.2: Symmetric and Hermitian matrices.

A matrix is symmetric if it is equal to its transpose. Symmetric (real-valued) matrices have the curious property that their eigenvalues are real and one can always find an orthonormal basis of eigenvectors. For complex-valued matrices, this is true as long as the matrix is "Hermitian", which means that it is equal to its conjugate transpose. After some examples and non-examples, we define define a self-adjoint linear map to be one such that (Av,w)=(v,Aw) always holds. This is the generalizion of both symmetric and Hermitian, and we will soon study these mappings in infinite-dimensional spaces of functions, and learn that these too have real eigenvalues and orthonormal eigenvectors. We conclude this lecture with several examples of "non-standard" inner products in R^n.

Course webpage (with lecture notes, homework, worksheets, etc.): http://www.math.clemson.edu/~macaule/...
Prerequisite: http://www.math.clemson.edu/~macaule/...