Let \((\mid)\) be the standard inner product on \(\RR^2\), and let \(T\) be the linear operator \(T(x_1, x_2) = (-x_2, x_1)\). Now \(T\) is ‘rotation through \(90^\circ\)’ and has the property that \((\alpha \mid T\alpha) = 0\) for all \(\alpha\) in \(\RR^2\). Find all inner products \([\mid]\) on \(\RR^2\) such that \([\alpha \mid T\alpha] = 0\) for each \(\alpha\).
Solution
1. Matrix representation of an inner product on \(\mathbb{R}^2\):
1.1. Any inner product \([\cdot \mid \cdot]\) on \(\mathbb{R}^2\) is uniquely represented with respect to the standard basis \(\{e_1, e_2\}\) by a symmetric, positive-definite matrix \(A = \begin{pmatrix} a & b \\ b & c \end{pmatrix} \in M_2(\mathbb{R})\):\[ [\alpha \mid \beta] = \alpha^t A \beta = ax_1 y_1 + b(x_1 y_2 + x_2 y_1) + cx_2 y_2, \quad \text{for } \alpha = \begin{pmatrix} x_1 \\ x_2 \end{pmatrix}, \beta = \begin{pmatrix} y_1 \\ y_2 \end{pmatrix}. \]
Riesz representation for bilinear forms on finite-dimensional spaces.
1.2. Positive-definiteness of \(A\) requires \(a > 0\), \(c > 0\), and \(\det(A) = ac - b^2 > 0\).
Sylvester’s criterion for \(2 \times 2\) real symmetric matrices.
2. Express the condition \([\alpha \mid T\alpha] = 0\):
2.1. For \(\alpha = \begin{pmatrix} x_1 \\ x_2 \end{pmatrix}\), \(T\alpha = \begin{pmatrix} -x_2 \\ x_1 \end{pmatrix}\).
definition of \(T\).
2.2. Compute \([\alpha \mid T\alpha]\):\[ [\alpha \mid T\alpha] = \begin{pmatrix} x_1 & x_2 \end{pmatrix} \begin{pmatrix} a & b \\ b & c \end{pmatrix} \begin{pmatrix} -x_2 \\ x_1 \end{pmatrix} = \begin{pmatrix} x_1 & x_2 \end{pmatrix} \begin{pmatrix} bx_1 - ax_2 \\ cx_1 - bx_2 \end{pmatrix} = b x_1^2 + (c - a) x_1 x_2 - b x_2^2. \]
matrix multiplication.
3. Determine the constraints on \(a, b, c\):
3.1. The quadratic form \(Q(x_1, x_2) = b x_1^2 + (c - a) x_1 x_2 - b x_2^2\) must vanish identically for all \((x_1, x_2) \in \mathbb{R}^2\).
hypothesis \([\alpha \mid T\alpha] = 0\) for all \(\alpha\).
3.2. Setting \((x_1, x_2) = (1, 0)\) gives \(Q(1, 0) = b = 0\).
evaluation.
3.3. Setting \((x_1, x_2) = (1, 1)\) with \(b = 0\) gives \(Q(1, 1) = c - a = 0 \implies c = a\).
evaluation.
3.4. Thus the matrix \(A\) must be of the form \(A = \begin{pmatrix} a & 0 \\ 0 & a \end{pmatrix} = a I_2\).
3.3.
3.5. By positive-definiteness, \(a > 0\).
1.
4. Conclusion: The inner products \([\cdot \mid \cdot]\) on \(\mathbb{R}^2\) satisfying \([\alpha \mid T\alpha] = 0\) for all \(\alpha\) are precisely the positive scalar multiples of the standard inner product:\[ [\alpha \mid \beta] = c (\alpha \mid \beta) \quad \text{for some constant } c > 0. \]
2 and 3.
Q.E.D.