Exercise 3. A matrix A E NA ( HR ) is said to be positive definite if U' AU > O for cach non - 2010 Vector UE THEM. ( 1) Show that the set of positive - definite matrices is closed under addition , and under scalar multiplication by positive scalars . ( The geometric picture here is that the set. of positive - definite matrices , while not a subspace , is a " conc ! ) ( ii ) Show that a diagonal matrix A is positive definite if and only if all of its diagonal entries are positive . ( jjj ) Given a symmetric , positive definite matrix A , show that ( V , W ) A : = UT Aw defines all inner product on Them." ( iv ) Show that , conversely , every inner product on {~ is of the above form , for some symmetric , positive definite matrix A