Positive Definite Quadratic Form

A Quadratic Form $Q({\bf x})$ is said to be positive definite if $Q({\bf x})>0$ for ${\bf x}\not={\bf0}$. A Real Quadratic Form in $n$ variables is positive definite Iff its canonical form is

$$Q({\bf z})={z_1}^2+{z_2}^2+\ldots+{z_n}^2.$$ (1)

A Binary Quadratic Form

$$F(x,y)=a_{11}x^2+2a_{12}xy+a_{22}y^2$$ (2)

of two Real variables is positive definite if it is $>0$ for any $(x,y)\not=(0,0)$, therefore if $a_{11}>0$ and the Discriminant $a\equiv a_{11}a_{22}-{a_{12}}^2>0$. A Binary Quadratic Form is positive definite if there exist Nonzero $x$ and $y$ such that

$$(ax^2+2bxy+cy^2)^2\leq {\textstyle{4\over 3}}\vert ac-b^2\vert$$ (3)

(Le Lionnais 1983).

A Quadratic Form $({\bf x}, {\hbox{\sf A}}{\bf x})$ is positive definite Iff every Eigenvalue of ${\hbox{\sf A}}$ is Positive. A Quadratic Form $Q=({\bf x}, {\hbox{\sf A}}{\bf x})$ with ${\hbox{\sf A}}$ a Hermitian Matrix is positive definite if all the principal minors in the top-left corner of ${\hbox{\sf A}}$ are Positive, in other words

$$a_{11} > 0$$ (4)

$$\left\vert\begin{array}{cc}a_{11} & a_{12}\\ a_{21} & a_{22}\end{array}\right\vert > 0$$ (5)

$$\left\vert\begin{array}{ccc}a_{11} & a_{12} & a_{13}\\ a_{21} & a_{22} & a_{23}\\ a_{31} & a_{32} & a_{33}\end{array}\right\vert > 0.$$ (6)

See also Indefinite Quadratic Form, Positive Semidefinite Quadratic Form

References

Gradshteyn, I. S. and Ryzhik, I. M. Tables of Integrals, Series, and Products, 5th ed. San Diego, CA: Academic Press, p. 1106, 1979.

Le Lionnais, F. Les nombres remarquables. Paris: Hermann, p. 38, 1983.