For any topic related to matrices. This includes: systems of linear equations, eigenvalues and eigenvectors (diagonalization, triangularization), determinant, trace, characteristic polynomial, adjugate, transpose, Jordan normal form, matrix algorithms (e.g. LU, Gauss elimination, SVD, QR), ...
Let $H$ be a symmetric matrix. Show that if $u$ and $v$ are eigenvectors of $H$ corresponding to distinct eigenvalues, then $u$ and $v$ are $H$-conjugate. Need some help, what and how should I ...