Sylvestrovo kritérium

Nech A je symetrická matica, kde

${\displaystyle A=\left({\begin{array}{cccc}a_{11}&a_{12}&\ldots &a_{1n}\\a_{21}&a_{22}&\ldots &a_{2n}\\\vdots &\vdots &\ddots &\vdots \\a_{n1}&a_{n2}&\ldots &a_{nn}\end{array}}\right)}$.

Označme ${\displaystyle D_{1},D_{2},\ldots ,D_{n}}$ determinanty definované nasledovne:

${\displaystyle D_{1}=|a_{11}|,\quad D_{2}=\left|{\begin{array}{cc}a_{11}&a_{12}\\a_{21}&a_{22}\end{array}}\right|,\quad \ldots ,\quad D_{n}=\left|{\begin{array}{cccc}a_{11}&a_{12}&\ldots &a_{1n}\\a_{21}&a_{22}&\ldots &a_{2n}\\\vdots &\vdots &\ddots &\vdots \\a_{n1}&a_{n2}&\ldots &a_{nn}\end{array}}\right|=|A|.}$

Potom matica A je kladne definitná práve vtedy, keď sú všetky determinanty ${\displaystyle D_{1},D_{2},\ldots ,D_{n}}$ kladné.