A quadratic form $f: \R^n \to \R $ is a function
$$f(x) = \transpose{ x }Qx $$
where $Q $ is an $n \times n$ real matrix. There is no loss of generality in assuming $Q$ to be symmetric, that is, $Q = \transpose{ Q }$. For if the matrix $Q $ is not symmetric, we can always replace it with the symmetric matrix.
$$Q_0 = \transpose{ Q_0 } = \frac{ 1 }{ 2 } (Q + \transpose{ Q }) $$
Note that
$$\transpose{ x } Q x = \transpose{ x } Q_0 x = \transpose{ x } (\frac{ 1 }{ 2 }Q + \frac{ 1 }{ 2 } \transpose{ Q })x $$
A quadratic form $\transpose{ \b{x}} \b{Q} \b{x}, \b{Q} = \transpose{ \b{Q}} $, is said to be
- positive definite if $\forall \b{x}, \transpose{ \b{x}} \b{Q} \b{x} > 0 $;
- positive semidefinite if $\forall \b{x}, \transpose{ \b{x}} \b{Q} \b{x} \geq 0$;
- negative definite if $\forall \b{x}, \transpose{ \b{x}} \b{Q} \b{x} < 0$;
- negative semidefinite if $\forall \b{x}, \transpose{ \b{x}} \b{Q} \b{x} \leq 0$.
The principle minors are $\det \b{Q} $ itself and the determinants of matrices obtained by successively removing an $i$th row and an $i$th column. That is, the principal minors are:
$$\det {\Mqr{{\vcr{ q_{i_1i_1}}{ q_{i_2i_1}}{ \vdots }{ q_{i_pi_1}}}}{{\vcr{ q_{i_1i_2}}{ q_{i_2i_2}}{ \vdots }{q_{i_pi_1}}}}{{\vcr{ … }{ … }{ }{ … }}}{{\vcr{ q_{i_1i_p}}{ q_{i_2i_p}}{ \vdots }{q_{i_pi_p}}}}}\ \ , p < n, \forall 1 \leq k \leq p, 1 \leq i_k \leq n $$
The leading principal minors are $\det \b{Q} $ and the minors obtained by successively removing the last row and the last column. That is, the leading principal minors are:
$$\Delta_1 = q_{11}, \Delta_2 = \det {\Mww{ q_{11}}{ q_{12}}{ q_{21}}{ q_{22}}}, \Delta_3 = \det {\Mee{ q_{11}}{ q_{12}}{ q_{13}}{ q_{21}}{ q_{22}}{ q_{23}}{ q_{31}}{ q_{32}}{ q_{33}}}, … , \Delta_n = \det \b{Q} $$
A quadratic form $\transpose{\b{x}}\b{Q}\b{x} = \transpose{\b{Q}}$ is positive definite
$\iff$ the leading principal minors of $\b{Q}$ are positive.