This paper focuses on the study of the second-order directional derivative of a symmetric matrix-valued function of the form F (X) = Pdiag[f(λ1(X)), · · · , f(λn(X))]P . For this purpose, we first adopt a direct way to derive the formula for the second-order directional derivative of any eigenvalue of a matrix in Torki [13]; Second, we establish a formula for the (parabolic) second-order direct...

Computing the fundamental matrix is the first step of many computer vision applications including camera calibration, image rectification and structure from motion. A new method for the estimation of the fundamental matrix from point correspondences is presented. The minimization of the geometric error is performed based Linfinity norm minimization framework. A single global minimum exists and ...

In this paper we complement our recent result on the explicit formula for the planar limit of the free energy of the two–matrix model by computing the second and third order observables of the model in terms of canonical structures of the underlying genus g spectral curve. In particular we provide explicit formulas for any three–loop correlator of the model. Some explicit examples are worked out.

We propose a generic method for iteratively approximating various second-order gradient steps - Newton, Gauss-Newton, Levenberg-Marquardt, and natural gradient - in linear time per iteration, using special curvature matrix-vector products that can be computed in O(n). Two recent acceleration techniques for on-line learning, matrix momentum and stochastic meta-descent (SMD), implement this appro...

We consider the second order matrix differential systems (1) (P(t)Y1)'+ Q(t)Y = 0 and (2) Y" + Q(t)Y = 0 where Y, P , and Q are n x n real continuous matrix functions with P(t) , Q(t) symmetric and P(t) positive definite for t E [to, cc) (P(t) > 0 , t > to) . We establish sufficient conditions in order that all prepared solutions Y(t) of (1) and (2) are oscillatory. The results obtained can be ...

Besides the fact that – by definition – matrix-exponential processes (MEPs) are more general than Markovian arrival processes (MAPs), only very little is known about the precise relationship of these processes in matrix notation. For the first time, this paper proves the persistent conjecture that – in two dimensions – the respective sets, MAP(2) and MEP(2), are indeed identical with respect to...

on [0,∞), where Y (t), P (t) and Q(t) are n × n real continuous matrix functions on [0,∞) with P (t), Q(t) symmetric and P (t) positive definite for t ∈ [0,∞) (P (t) > 0, t ≥ 0). A solution Y (t) of (1.1) is said to be nontrivial if det Y (t) 6= 0 for at least one t ∈ [0,∞) and a nontrivial solution Y (t) of (1.1) is said to be prepared (selfconjugated) if Y ∗(t)P (t)Y ′(t)− Y ∗′(t)P (t)Y (t) ≡...

In this note, we consider a generalized Fibonacci sequence fung : Then give a generating matrix for the terms of sequence fukng for a positive integer k: With the aid of this matrix, we derive some new combinatorial identites for the sequence fukng.

We develop a method that allows us to construct families of orthogonal matrix polynomials of size N ×N satisfying second order difference equations with polynomial coefficients. The existence (and properties) of these orthogonal families strongly depends on the non commutativity of the matrix product, the existence of singular matrices and the matrix size N .