Exercise 1 lists the following quadratic forms \(q(x) = q(x_1, x_2)\) on \(\mathbb{R}^2\) and asks for the symmetric bilinear form \(f\) corresponding to each:
\(q(x) = a x_1^2\);
\(q(x) = b x_1 x_2\);
\(q(x) = c x_2^2\);
\(q(x) = 2 x_1^2 - \frac{1}{3} x_1 x_2\);
\(q(x) = x_1^2 + 9 x_2^2\);
\(q(x) = 3 x_1 x_2 - x_2^2\);
\(q(x) = 4 x_1^2 + 6 x_1 x_2 - 3 x_2^2\).
Find the matrix, in the standard ordered basis, and the rank of each of the bilinear forms determined in Exercise 1. Indicate which forms are non-degenerate.
Solution
For a quadratic form \[ q(x_1,x_2)=A x_1^2+2B x_1x_2+C x_2^2, \] the associated symmetric bilinear form has standard matrix \[ M= \begin{pmatrix} A&B\\ B&C \end{pmatrix}. \] On \(\mathbb R^2\), the form is non-degenerate exactly when \(\det M\ne0\), equivalently when \(M\) has rank two.
1. Part (a): \(q=a x_1^2\).
The matrix is \[ M_a= \begin{pmatrix} a&0\\ 0&0 \end{pmatrix}. \] Thus \[ \operatorname{rank}M_a =\begin{cases} 1,&a\ne0,\\ 0,&a=0. \end{cases} \] In either case the form is degenerate.
2. Part (b): \(q=b x_1x_2\).
Here \[ M_b= \begin{pmatrix} 0&b/2\\ b/2&0 \end{pmatrix}, \qquad \det M_b=-\frac{b^2}{4}. \] Hence \[ \operatorname{rank}M_b =\begin{cases} 2,&b\ne0,\\ 0,&b=0, \end{cases} \] and the form is non-degenerate exactly when \(b\ne0\).
3. Part (c): \(q=c x_2^2\).
The matrix is \[ M_c= \begin{pmatrix} 0&0\\ 0&c \end{pmatrix}. \] Thus \[ \operatorname{rank}M_c =\begin{cases} 1,&c\ne0,\\ 0,&c=0, \end{cases} \] and the form is always degenerate.
4. Part (d): \(q=2x_1^2-\frac13x_1x_2\).
The matrix is \[ M_d= \begin{pmatrix} 2&-1/6\\ -1/6&0 \end{pmatrix}, \] and \[ \det M_d=-\frac1{36}\ne0. \] Therefore \[ \operatorname{rank}M_d=2, \] so the form is non-degenerate.
5. Part (e): \(q=x_1^2+9x_2^2\).
The matrix is \[ M_e= \begin{pmatrix} 1&0\\ 0&9 \end{pmatrix}, \] with determinant \(9\). Hence its rank is two and the form is non-degenerate.
6. Part (f): \(q=3x_1x_2-x_2^2\).
The matrix is \[ M_f= \begin{pmatrix} 0&3/2\\ 3/2&-1 \end{pmatrix}, \] and \[ \det M_f=-\frac94\ne0. \] Thus its rank is two and the form is non-degenerate.
7. Part (g): \(q=4x_1^2+6x_1x_2-3x_2^2\).
The matrix is \[ M_g= \begin{pmatrix} 4&3\\ 3&-3 \end{pmatrix}, \] with \[ \det M_g=4(-3)-9=-21\ne0. \] Therefore its rank is two and the form is non-degenerate.
In summary, (d), (e), (f), and (g) are always non-degenerate; (b) is non-degenerate exactly when \(b\ne0\); and (a), (c) are always degenerate.