Indefinite Semidefinite and Definite Binary Quadratic Forms

Indefinite Semidefinite and Definite Binary Quadratic Forms

Definition: Let $f(x, y) = ax^2 + bxy + cy^2$ be a binary quadratic form.
1) $f(x, y)$ is Indefinite if $f$ takes on both positive and negative values.
2) $f(x, y)$ is Positive Semidefinite if $f(x, y) \geq 0$ for all integers $x, y \in \mathbb{Z}$.
3) $f(x, y)$ is Negative Semidefinite if $f(x, y) \leq 0$ for all integers $x, y \in \mathbb{Z}$.
4) $f(x, y)$ is Semidefinite if it is either positive semidefinite or negative semidefinite.
5) $f(x, y)$ is Positive Definite if $f(x, y) \geq 0$ for all integers $x, y \in \mathbb{Z}$ and $f(x, y) = 0$ if and only if $(x, y) = (0, 0)$.
6) $f(x, y)$ is Negative Definite if $f(x, y) \leq 0$ for all integers $x, y \in \mathbb{Z}$ and $f(x, y) = 0$ if and only if $(x, y) = (0, 0)$.
7) $f(x, y)$ is Definite if it is either positive definite or negative definite.
  • The following BQF is indefinite:
(1)
\begin{equation} f(x, y) = x^2 + xy - y^2 \end{equation}
  • This is because $f(1, 1) = 1 > 0$, but $f(-1, 1) = -1 < 0$.
  • The following BQF is positive semidefinite:
(2)
\begin{equation} f(x, y) = x^2 + 2xy + y^2 \end{equation}
  • This is because $f(x, y) = (x + y)^2 \geq 0$ for all $x, y \in \mathbb{Z}$. Observe that $f(x, y) = 0$ whenever $x = -y$.
  • The following BQF is negative semidefinite:
(3)
\begin{align} \quad f(x, y) = -x^2 - 2xy - y^2 \end{align}
  • This is because $f(x, y) = - (x + y)^2 \leq 0$ for all $x, y \in \mathbb{Z}$. Observe that $f(x, y) = 0$ whenever $x = -y$.
  • The following BQF is positive definite:
(4)
\begin{align} \quad f(x, y) = x^2 + y^2 \end{align}
  • This is because $f(x, y) \geq 0$ for all $x, y \in \mathbb{Z}$ and that $f(x, y) = 0$ if and only if $x = 0$ and $y = 0$.
  • The following BQF is negative definite:
(5)
\begin{align} \quad f(x, y) = -x^2 - y^2 \end{align}
  • This is because $f(x, y) \leq 0$ for all $x, y \in \mathbb{Z}$ and that $f(x, y) = 0$ if and only if $x = 0$ and $y = 0$.
Theorem 1: Let $f(x, y)$ be a binary quadratic form.
a) If $d > 0$ then $f(x, y)$ is indefinite.
b) If $d = 0$ then $f(x, y)$ is semidefinite but not definite.
c) If $d < 0$ then $f(x, y)$ is either definite. In particular, $f(x, y)$ is positive definite if both $a, c > 0$ or negative definite if both $a, c < 0$.
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