# Is a symmetric, invertible and positive semi-definite matrix automatically positive definite?

My question is: Is a symmetric, invertible and positive semi-definite matrix automatically positive definite?

Any solution or hints on how this can be answered would be appreciated.

• Hint: if any of the principal minors is $0$, then there exists a nonzero vector $x$ with $x^T Ax =0$ (why?). – darij grinberg May 17 '18 at 22:38

Note that for a positive semi-definite matrix ($\lambda_i\ge 0$) and invertible ($\lambda_i\neq 0$) we have that $\lambda_i> 0$ then the matrix is positive definite.
• Do you have any reference where i can connect the notions that Positive (semi-)definite is equivalent to $\lambda_i (\geq) >0$? – Martin May 17 '18 at 19:21
• @Martin Symply recall that a symmetric matrix is diagonalizable, then $x^TAx$ with respect to a basis of orthogonal eigenvectors becomes $$y^TM^TAMy=y^TDy=\sum \lambda_iy_i^2$$ – gimusi May 17 '18 at 19:28