“positive matrices” in Sylvester's criterion

From Wikipedia:

Sylvester's criterion states that a Hermitian matrix ''M'' is positive-definite if and only if all the following matrices have a positive determinant:

• the upper left 1-by-1 corner of $M$,
• the upper left 2-by-2 corner of $M$,
• the upper left 3-by-3 corner of $M$,
• ...
• $M$ itself.

In other words, all of the leading principal minors must be positive.

So do we call a principal minor positive if it has a positive determinant? Because from my knowledge a positive matrix is just a matrix with positive elements. In this sense the positive definite hermitian (symmetric) matrix

$$\begin{pmatrix} 2 & -1 \\ -1 & 2 \end{pmatrix}$$

would have principal minors which are not positive matrices.

• The word minor usually refers to a subdeterminant (not a submatrix). – Hans Lundmark May 11 '15 at 15:15

In the usage of this context, the $m \times m$ upper-left corner of a matrix is referred to as the leading principal submatrix.
The determinant of this matrix is referred to as the $m$th leading principal minor.
Positive-definite is not positive elements. It means that for every vector $x\not=0$: $$x^TAx\geq0.$$ So, yes, this hermitian matrix is possitive.
• Your notations seem off. Do you mean that $x^T A x \geq 0$? – Olivier Mar 24 '18 at 2:06