Positive Definite Proof

[renolovexoxo]

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.

 

[HallsofIvy]

That looks pretty straight forward to me. What do you get if you put A+ B into the definition of "positive definite"- and use the fact that a dot product is linear?

 

[renolovexoxo]

Part b is really what is giving me more trouble.