This is an introduction to the notion of numerical range for bounded linear operators on Hilbert space. The main results are: determination of the numerical range for two by two matrices, the Toeplitz-Hausdorff Theorem establishing the convexity of the numerical range for any Hilbert space operator, and a detailed discussion of the relationship between the numerical range and the spectrum. The ...

in this paper, we introduce the new class of implicit l-stable generalized hybrid methods forthe numerical solution of first order initial value problems. we generalize the hybrid methodswith utilize ynv directly in the right hand side of classical hybrid methods. the numericalexperimentation showed that our method is considerably more efficient compared to wellknown methods used for the numer...

We analyze the numerical range of high-dimensional random matrices, obtaining limit results and corresponding quantitative estimates in the non-limit case. For a large class of random matrices their numerical range is shown to converge to a disc. In particular, numerical range of complex Ginibre matrix almost surely converges to the disk of radius √ 2. Since the spectrum of non-hermitian random...

We consider the problem of finding the optimal correlation between two projected matrices U∗AU and V ∗BV . The square matrices A and B may be of different dimensions, but the isometries U and V have a common column dimension k. The correlation is measured by the real function c(U,V ) = R tr(U∗AUV ∗B∗V ), which we maximize of the isometries U∗U = V ∗V = Ik. This problem can be viewed as an exten...

In this paper we derive a successive matrix squaring (SMS) algorithm to approximate an outer generalized inverse with prescribed range and null space of a given matrix A ∈ Cm×n r . We generalize the results from the papers [3], [16], [18], and obtain an algorithm for computing various classes of outer generalized inverses of A. Instead of particular matrices used in these articles, we use an ap...