The question is to prove that positive definite matrix is invertible. I've proved that positive definite matrix has null space of zero vector. So actually we can prove that a matrix has an null space of zero vector is invertible. How to prove this? I need a basic way of proof without using other conclusions (like all eigenvalues are positive so it's invertible or a positive definite matrix is nonsingular so its inverse exists) used in the proving process. Thank you!
[Math] prove that positive definite matrix is invertible
linear algebra
Best Answer
Now that you know that $A\in M_n$ it is injective, use the rank-nullity theorem to deduce that $$ \mbox{rank}A=n-\mbox{null}A=n. $$ It follows that $A$ is also surjective, hence invertible.