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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.

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

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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.

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  • $\begingroup$ Your notations seem off. Do you mean that $x^T A x \geq 0$? $\endgroup$ – Olivier Mar 24 '18 at 2:06
  • $\begingroup$ Ok, thanks for the correction. $\endgroup$ – Ernesto Iglesias Mar 24 '18 at 2:21
  • $\begingroup$ In some books, a positive matrix is defined as a matrix with positive entries. So it is best to always check the definition. $\endgroup$ – AnyAD Mar 24 '18 at 3:22

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