The following expressions define quadratic forms \(q(x) = q(x_1, x_2)\) on \(\mathbb{R}^2\). Find the symmetric bilinear form \(f(x, y)\) corresponding to each quadratic form \(q\):
- \(q(x) = a x_1^2\)
- \(q(x) = b x_1 x_2\)
- \(q(x) = c x_2^2\)
- \(q(x) = 2 x_1^2 - \frac{1}{3} x_1 x_2\)
- \(q(x) = x_1^2 + 9 x_2^2\)
- \(q(x) = 3 x_1 x_2 - x_2^2\)
- \(q(x) = 4 x_1^2 + 6 x_1 x_2 - 3 x_2^2\)
Solution
Goal: Compute the unique symmetric bilinear form \(f(x, y)\) from a quadratic form \(q(x)\) on \(\mathbb{R}^2\) using polarization: \(f(x, y) = \frac{1}{2}[q(x + y) - q(x) - q(y)]\).
1. Polarization Identity on \(\mathbb{R}^2\):Proof:
1.1. A general quadratic form on \(\mathbb{R}^2\) is given by:\[q(x) = A x_1^2 + 2B x_1 x_2 + C x_2^2 = x^T \begin{bmatrix} A & B \\ B & C \end{bmatrix} x.\]
1.2. The unique symmetric bilinear form \(f: \mathbb{R}^2 \times \mathbb{R}^2 \to \mathbb{R}\) satisfying \(f(x, x) = q(x)\) is given by:\[f(x, y) = x^T \begin{bmatrix} A & B \\ B & C \end{bmatrix} y = A x_1 y_1 + B(x_1 y_2 + x_2 y_1) + C x_2 y_2.\]
1.3. In particular, pure quadratic terms \(x_i^2\) polarize to \(x_i y_i\), and cross terms \(x_1 x_2\) polarize to \(\frac{1}{2}(x_1 y_2 + x_2 y_1)\).
2. Computations for Each Part:Proof:
2.1. Part (a): \(q(x) = a x_1^2\)- Matrix: \(\begin{bmatrix} a & 0 \\ 0 & 0 \end{bmatrix}\).- Symmetric form: \(f(x, y) = a x_1 y_1\).
2.2. Part (b): \(q(x) = b x_1 x_2\)- Matrix: \(\begin{bmatrix} 0 & b/2 \\ b/2 & 0 \end{bmatrix}\).- Symmetric form: \(f(x, y) = \frac{b}{2}(x_1 y_2 + x_2 y_1)\).
2.3. Part (c): \(q(x) = c x_2^2\)- Matrix: \(\begin{bmatrix} 0 & 0 \\ 0 & c \end{bmatrix}\).- Symmetric form: \(f(x, y) = c x_2 y_2\).
2.4. Part (d): \(q(x) = 2 x_1^2 - \frac{1}{3} x_1 x_2\)- Matrix: \(\begin{bmatrix} 2 & -1/6 \\ -1/6 & 0 \end{bmatrix}\).- Symmetric form: \(f(x, y) = 2 x_1 y_1 - \frac{1}{6}(x_1 y_2 + x_2 y_1)\).
2.5. Part (e): \(q(x) = x_1^2 + 9 x_2^2\)- Matrix: \(\begin{bmatrix} 1 & 0 \\ 0 & 9 \end{bmatrix}\).- Symmetric form: \(f(x, y) = x_1 y_1 + 9 x_2 y_2\).
2.6. Part (f): \(q(x) = 3 x_1 x_2 - x_2^2\)- Matrix: \(\begin{bmatrix} 0 & 3/2 \\ 3/2 & -1 \end{bmatrix}\).- Symmetric form: \(f(x, y) = \frac{3}{2}(x_1 y_2 + x_2 y_1) - x_2 y_2\).
2.7. Part (g): \(q(x) = 4 x_1^2 + 6 x_1 x_2 - 3 x_2^2\)- Matrix: \(\begin{bmatrix} 4 & 3 \\ 3 & -3 \end{bmatrix}\).- Symmetric form: \(f(x, y) = 4 x_1 y_1 + 3(x_1 y_2 + x_2 y_1) - 3 x_2 y_2\).
3. Conclusion:The symmetric bilinear forms are obtained by polarizing \(x_i^2 \mapsto x_i y_i\) and \(x_1 x_2 \mapsto \frac{1}{2}(x_1 y_2 + x_2 y_1)\). Q.E.D.