4. For a unitary operator U, show that T = -ilog(U) is Hermitian and therefore thatU = eil for some Hermitian T.

Answered: -i log(U) is Hermitian and therefore…

Linear Algebra: A Modern Introduction

4th Edition

ISBN:9781285463247

Author:David Poole

Publisher:David Poole

Chapter7: Distance And Approximation

Section7.2: Norms And Distance Functions

Problem 33EQ

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4. For a unitary operator U, show that T = -ilog(U) is Hermitian and therefore that
U = eil for some Hermitian T.

Expert Solution

Step 1

Let U be a unitary operator.

To prove that $T = - i \log (U)$ is Hermitian.

Since, U is an unitary operator, $U U^{*} = U^{*} U = I$ where $U^{*}$ is the adjoint of U and $I$ is the identity operator.

As $T = - i \log (U)$ , therefore, it's adjoint is $T^{*} = i \log (U^{*})$

Now,

$\begin{array}{rcl} T - T^{*} & = & - i \log (u) - [i \log (U^{*})] \\ = & - i \log (U) - i \log (U^{*}) \\ = & - i \log (U) - i \log (U^{- 1}) \\ = & i \log (U) + i \log (U) [\log (A^{- 1}) = - \log (A)] \\ = & 0 \\ T & = & T^{*} \end{array}$

Hence, T is Hermitian.

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