7.3 From the construction of the S-matrix, we had identified the interaction matrix as that which encodes the non-trivial reflection and lack of transparency off a potential. In this problem, we'll provide another interpretation of the optical theorem and find a constraint on the eigenvalues of the interaction matrix.

(a) Prove that any matrix \(\mathbb{A}\) can be expressed as the sum of two Hermitian matrices \(\mathbb{H}_{1}, \mathbb{H}_{2}\) as \(\mathbb{A}=\mathbb{H}_{1}+i \mathbb{H}_{2}\).

(b) Using this result, we can write the interaction matrix as \(\hat{\mathcal{M}}=\mathbb{X}+i \mathbb{Y}\), for two Hermitian matrices \(\mathbb{X}, \mathbb{Y}\). Call the eigenvalues of these matrices \(x_{n}\) and \(y_{n}\), respectively, for \(n\) that ranges over the dimension of the interaction matrix. Show that these eigenvalues lie on the circle in the \(\left(x_{n}, y_{n}\right)\) plane centered at \((0,1)\) with radius 1 . The representation of this circle is called an Argand diagram.

(c) Explicitly determine the eigenvalues for the interaction matrix \(\hat{\mathcal{M}}\) of the narrow potential barrier provided in Eq. (7.94). Do they live on the Argand circle? Can any point on the Argand circle be realized, for particular values of momentum?