Some Basic Linear Algebra
Posted: Fri Jan 01, 2016 3:10 pm
Let \( \displaystyle A \in \mathbb{M}_{n}(\mathbb{R}) \) a symmetric and positive definite matrix. Show that :
- the diagonal elements of \( \displaystyle A \) are positive.
- the eigenvalues of \( \displaystyle A \) are positive.
- the determinant of \( \displaystyle A \) is positive.
- the absolutely maximum element of \( \displaystyle A \), that is \( \displaystyle \max_{ 1 \leq i,j \leq n } |a_{ij}| \), is on the diagonal, that is \( \displaystyle \max_{ 1 \leq i,j \leq n } |a_{ij}| = a_{kk} , \) for some \( \displaystyle k \in \{ 1, \dots, n\} \).