Preliminary Results

Abstract

Let

$$B = diag({\mu _1},...{\mu _N})$$

be a diagonal matrix. Suppose that μk (k = 1,2,..., N) are distinct complex numbers in the open upper halfplane. It is evident that

$$\frac{1}{i}(B - {B^*}) = diag(2{\mathop{\rm Im}\nolimits} {\mu _1},...,2{\mathop{\rm Im}\nolimits} {\mu _N})$$

and

$$rank(B - {B^*}) = N$$

, where the adjoint matrix is taken with respect to the standard scalar product

$$({h_1},{h_2}) = i\sum\limits_{j,k = 1}^N {\frac{{h_1^j\overline {h_2^k} }}{{{\mu _j} - \overline {{\mu _k}} }}}$$

in the space H = C^N of vectors

$$h = \left( \begin{array}{l}{h^1} \\. \\. \\. \\{h^N} \\\end{array} \right),{h^k} \in C$$

.