We say a symmetric matrix A is positive definite if Ax dot x>0 for all x not zero, negative definite if Ax dot x<0 for all x not zero and positive semi-definite if Ax dot x >/= 0 for all x.

Show that if A and B are positive definite, then so is A+B

Show that if A is positive definite if and only if all its eigenvalues are positive.