Positive Definite of a Hessian Matrix

Question

What is the sufficient condition for a Hessian Matrix to be a positive definite? In the case of a real matrix $x^TAx>0$ makes the matrix $A$ positive definite but we can't go for multiplying terms and checking if it is positive or not. Is there any direct check on $A$ which is sufficient for it to be positive definite?

Answer 1

For a symmetric matrix we can refer to the following criteria

• find all the eigen values, if all $n$ eigenvalues are positive (in general difficult to be applied)

• Completing the squares (difficult to be applied but very effective when we can)

• By Descartes' rule of signs applied to the characteristic polynomial (better than the previous)

• Sylvester's criterion (the best in my opinion)