Let \(V\) be a finite-dimensional inner product space, and let \(W\) be a subspace of \(V\). Then \(V = W \oplus W^{\perp}\), that is, each \(\alpha\) in \(V\) is uniquely expressible in the form \(\alpha = \beta + \gamma\), with \(\beta\) in \(W\) and \(\gamma\) in \(W^{\perp}\). Define a linear operator \(U\) by \(U\alpha = \beta - \gamma\).
Prove that \(U\) is both self-adjoint and unitary.
If \(V\) is \(\RR^3\) with the standard inner product and \(W\) is the subspace spanned by \((1, 0, 1)\), find the matrix of \(U\) in the standard ordered basis.
Solution
Goal: Prove that the orthogonal reflection operator \(U = P_W - P_{W^\perp}\) is self-adjoint and unitary in (a), and compute its matrix representation in \(\mathbb{R}^3\) for \(W = \operatorname{span}\{(1, 0, 1)\}\) in (b).
1. Part (a): Self-adjointness and unitarity of \(U\).Proof:
1.1. Let \(P_W: V \to W\) and \(P_{W^\perp}: V \to W^\perp\) denote the orthogonal projections onto \(W\) and \(W^\perp\).
1.2. Since \(V = W \oplus W^\perp\), the identity operator is \(I = P_W + P_{W^\perp}\), so \(P_{W^\perp} = I - P_W\).
1.3. The reflection operator is given by\[U = P_W - P_{W^\perp} = P_W - (I - P_W) = 2P_W - I.\]
1.4. Self-adjointness: Since orthogonal projections are self-adjoint (\(P_W^* = P_W\)), we compute\[U^* = (2P_W - I)^* = 2P_W^* - I^* = 2P_W - I = U.\]Thus \(U\) is self-adjoint.
1.5. Unitarity: Compute \(U^2\) using the idempotence \(P_W^2 = P_W\):\[U^2 = (2P_W - I)^2 = 4P_W^2 - 4P_W + I^2 = 4P_W - 4P_W + I = I.\]
1.6. Combining self-adjointness and the involution property \(U^2 = I\):\[U^* U = U U = U^2 = I \quad \text{and} \quad U U^* = U^2 = I.\]
1.7. Therefore \(U\) is unitary (and orthogonal in the real case).
2. Part (b): Matrix of \(U\) on \(\mathbb{R}^3\) for \(W = \operatorname{span}\{(1, 0, 1)\}\).Proof:
2.1. Let \(w_1 = (1, 0, 1)\). The norm squared is \(\|w_1\|^2 = 1^2 + 0^2 + 1^2 = 2\).
2.2. The orthogonal projection of \(\alpha = (x, y, z) \in \mathbb{R}^3\) onto \(W\) is\[P_W(x, y, z) = \frac{\langle (x, y, z), (1, 0, 1) \rangle}{\|w_1\|^2} w_1 = \frac{x + z}{2} (1, 0, 1) = \left( \frac{x + z}{2}, 0, \frac{x + z}{2} \right).\]
2.3. Applying \(U = 2P_W - I\):\[U(x, y, z) = 2 \left( \frac{x + z}{2}, 0, \frac{x + z}{2} \right) - (x, y, z) = (x + z, 0, x + z) - (x, y, z) = (z, -y, x).\]
2.4. Evaluating \(U\) on the standard basis \(\{e_1, e_2, e_3\}\) of \(\mathbb{R}^3\):\[U(e_1) = U(1, 0, 0) = (0, 0, 1) = 0 e_1 + 0 e_2 + 1 e_3,\]\[U(e_2) = U(0, 1, 0) = (0, -1, 0) = 0 e_1 - 1 e_2 + 0 e_3,\]\[U(e_3) = U(0, 0, 1) = (1, 0, 0) = 1 e_1 + 0 e_2 + 0 e_3.\]
2.5. Writing the image vectors as columns yields the matrix of \(U\) with respect to the standard ordered basis:\[[U] = \begin{bmatrix} 0 & 0 & 1 \\ 0 & -1 & 0 \\ 1 & 0 & 0 \end{bmatrix}.\]
3. Conclusion:Proof:\(U\) is self-adjoint (\(U^* = U\)) and unitary (\(U^* U = I\)), and its standard matrix representation is \(\begin{bmatrix} 0 & 0 & 1 \\ 0 & -1 & 0 \\ 1 & 0 & 0 \end{bmatrix}\).