Question

Verify equation (11.25). Hint: Remember from Section 9 that the transpose conjugate (dagger) of a product of matrices is the product of the transpose conjugates in reverse order and that ${\mathit{U}}^{\mathbf{+}}\mathbf{=}\mathit{U}\mathbf{-}\mathbf{1}$. Also remember that we have assumed real eigenvalues, so D is a real diagonal matrix.

Short answer

$U^{-1}{M}^{+}U=U^{-1}MU\Rightarrow M^{+}=M$

Step-by-step solution

Given Information

$U^{-1}MU=D$

Hermitian Matrix

A Hermitian matrix is a square matrix whose conjugate transpose matrix is equal to it. A Hermitian matrix's non-diagonal entries are all complex integers.

Hermitian Conjugate

Take the Hermitian conjugate of the first equation.

$$\begin{gathered} {\left(U^{-1}MU\right)}^{+}=U^{+}{M}^{+}{\left(U^{+}\right)}^{-1} \\ =U^{-1}{M}^{+}{\left(U^{-1}\right)}^{-1}\left[U^{+}=U^{-1}\right] \\ =U^{-1}{M}^{+}U \end{gathered}$$

The eigenvalue of M is real and D is a real diagonal matrix.

$D^{+}=D$

Therefore,

$$\begin{gathered} =U^{-1}{M}^{+}U \\ =U^{-1}MU \\ \Rightarrow M^{+}=M^{} \end{gathered}$$

Hence, verified.