Numerical Range

Optimally rotated vectors

“…Introduction. The concept of cosine of an operator or a matrix was first introduced by Gustafson (see [2]). Given an operator on a Hilbert space, cosine of T is defined by…”

“…This parameter has important applications in numerical analysis as well as pure matrix and operator theory. See [2] for more information on the applications of µ(T ) to numerical techniques such as conjugate gradient and steepest descent methods. In recent years, many attempts have been made to compute or approximate cos T for operators on complex Hilbert spaces.…”

“…In recent years, many attempts have been made to compute or approximate cos T for operators on complex Hilbert spaces. In particular, computation and approximation of cos T for normal operators have been somewhat successful (see [2,3,4,5]). A vector f for which the inf in (1.1) is attained is called a maximally rotated vector for T (a maximally rotated vector is called an antieigenvector in our earlier papers).…”

“…Unfortunately, a variational approach analogous to Rayleigh-Ritz variational theory of eigenvectors is not successful for computing optimally rotated vectors. Nevertheless, Gustafson has found an Euler equation that satisfies these vectors (see [2]). His approach is based on the direct computation of the left-hand side of…”

Application of the Two Nonzero Component Lemma in Resource Allocation

“…Second, the operational calculus method is futile in extending Kantorovich-type inequalities to operators on infinite dimensional Hilbert spaces. See [1] for examples using each of the two methods mentioned above. In his study of matrix inequalities and matrix optimization, the author has discovered and proved a lemma called the Two Nonzero Component Lemma or TNCL for short.…”