Question
- Consider the linear operator T : C3 → C3, defined by
T (z1, z2, z3) = (z1 − iz2, iz1 + 2z2 + iz3, −iz2 + z3) .
- Compute T ∗ and check whether T is self-adjoint.
ii) Check whether T is unitary
Answer :
Word Count : 548
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Skip in 10
To solve the problem, let's break it down step-by-step: We are given a linear operator \( T: \mathbb{C}^3 \to \mathbb{C}^3 \), defined by: \[ T(z_1, z_2, z_3) = \left( z_1 - i z_2, i z_1 + 2z_2 + i z_3, -i z_2 + z_3 \right) \] ### i) Compute \( T^* \) and check if \( T \) is self-adjoint. To compute the adjoint \( T^* \) of the operator \( T \), we need to find the Hermitian conjugate of each component of \( T \). The adjoint operator \( T^* \) is defined as the operator that satisfies: \[ \langle T\mathbf{v}, \mathbf{w} \rangle = \langle \mathbf{v}, T^* \mathbf{w} \rangle \] Where \( \langle \cdot, \cdot \rangle \) is the standard Hermitian inner product for complex vectors. Specifically, \( \langle (a_1, a_2, a_3), (b_1, b_2, b_3) \rangle = \overline{a_1} b_1 + \overline{a_2} b_2 + \overline{a_3} b_3 \). We now compute \( T^*
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- Compute T ∗ and check whether T is self-adjoint.
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