Exercise 3. A matrix A E NA ( HR ) is said to be positive definite if U' AU gt; O for cach non - 2010 Vector UE THEM. ( 1) Show that the set of...

Question in the picture.

Don't copy the answer from chegg, thanks.

MATH 236 in McGill, Algebra2

Exercise 3. A matrix A E NA ( HR ) is said to be positive definite if U' AU > O for cachnon - 2010 Vector UE THEM.( 1) Show that the set of positive - definite matrices is closed under addition , and underscalar 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 diagonalentries are positive .( jjj ) Given a symmetric , positive definite matrix A , show that ( V , W ) A : = UT Aw definesall inner product on Them."( iv ) Show that , conversely , every inner product on {~ is of the above form , for somesymmetric , positive definite matrix A.