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)

\begin{align} \quad f(x, y) = g(m_{11}x + m_{12}y, m_{21}x + m_{22}y) \end{align}

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)

\begin{align} \quad f(x, y) = f(m_{11}x + m_{12}y, m_{21}x + m_{22}y) \end{align}

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)

\begin{align} \quad f(n_{11}x + n_{12}y, n_{21}x + n_{22}y) &= g(m_{11}[n_{11}x + n_{12}y] + m_{12}[n_{21}x + n_{22}y], m_{21}[n_{11}x + n_{12}y] + m_{22}[n_{21}x + n_{22}y]) \\ &= g([m_{11}m_{22} - m_{12}m_{21}]x + [-m_{11}m_{12} + m_{12}m_{11}]y, [m_{21}m_{22} - m_{22}m_{21}]x + [-m_{21}m_{12} + m_{22}m_{11}]x) \\ &= g(x, y) \end{align}

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)

\begin{align} \quad f(m_{11}x + m_{12}y, m_{21}x + m_{22}y) &= f(3x + 2y, x + y) \\ &= (3x + 2y)^2 + 3(3x + 2y)(x + y) + 4(x + y)^2 \\ &= 9x^2 + 12xy + 4y^2 + 3[3x^2 + 5xy + 2y^2] + 4[x^2 + 2xy + y^2] \\ &= 9x^2 + 12xy + 4y^2 + 9x^2 + 15xy + 6y^2 + 4x^2 + 8xy + 4y^2 \\ &= 22x^2 + 35xy + 14y^2 \end{align}