Matricial inner products and pointed cones of Hermitian-preserving linear transformations
Let \(V\) be a vector space over the real or complex numbers. A subset \(K\) of \(V\) is called a cone if for \(x,y\) in \(K\) and \(\alpha \geq 0\) the vector \(\alpha x+y\) is also in \(K\). Let \(M_n\) be the set of \(n\times n\) matrices with real or complex entries. The authors introduce a matricial inner product as a function \(M_n \times M_n \to M_n\), \((A,B) \to B^*PA+ AQB^*\), where \(P\) and \(Q\) are positive semidefinite and at least one is positive definite. They show that this function is a vectorial inner product with respect to the reflector \(\sigma\), where \(\sigma (A) =A^*\). The matricial inner product is used to show that certain sets of matrices are cones.
scientific article; zbMATH DE number 1011323
scientific article