Table of Contents

Example Problems Regarding Equivalent Binary Quadratic Forms
Recall from the Equivalent Binary Quadratic Forms page that if $f$ and $g$ are binary quadratic forms then $f$ is said to be equivalent to $g$ if there exists $m_{11}, m_{12}, m_{21}, m_{22} \in \mathbb{Z}$ such that:
(1)and $m_{11}m_{22}  m_{12}m_{21} = 1$. We will now look at some practice problems regarding equivalent binary quadratic forms.
Example 1
On the page above, we proved that equivalence of binary quadratic forms is indeed an equivalence relation. Prove that $\sim$ is reflexive and symmetric directly.
First we show that $\sim$ is reflexive. Let $m_{11} = 1$, $m_{12} = 0$, $m_{21} = 0$, and $m_{22} = 1$. Then:
(2)and $m_{11}m_{22}  m_{12}m_{22} = 1$. So $f \sim f$ and $\sim$ is reflexive.
Second we show that $\sim$ is symmetric. Suppose that $f \sim g$. Then $f(x, y) = g(m_{11}x + m_{12}y, m_{21}x + m_{22}y)$ where $m_{11}m_{22}  m_{12}m_{21} = 1$. Let $n_{11} = m_{22}$, $n_{12} = m_{12}$, $n_{21} = m_{21}$, and $n_{22} = m_{11}$. Then:
(3)So $g \sim f$ and $\sim$ is symmetric.
Example 2
Let $f(x, y) = x^2 + 3xy + 4y^2$. Use the matrix $M = \begin{bmatrix} 3 & 2 \\ 1 & 1 \end{bmatrix}$ to obtain an equivalent binary quadratic form $g(x, y)$.
We have that $m_{11} = 3$, $m_{12} = 2$, $m_{21} = 1$, and $m_{22} = 1$, so:
(4)