We consider the matrix completion problem of recovering a structured matrix from noisy and partial measurements. Recent works have proposed tractable estimators with strong statistical guarantees for the case where the underlying matrix is low–rank, and the measurements consist of a subset, either of the exact individual entries, or of the entries perturbed by additive Gaussian noise, which is ...

The exponential map is important because it provides a map from the Lie algebra of a Lie group into the group itself. We focus on matrix groups over the quaternions and the exponential map from their Lie algebras into the groups. Since quaternionic multiplication is not commutative, the process of calculating the exponential of a matrix over the quaternions is more involved than the process of ...

The Lanczos method is an iterative procedure to compute an orthogonal basis for the Krylov subspace generated by a symmetric matrix A and a starting vector v. An interesting application of this method is the computation of the matrix exponential exp(−τA)v. This vector plays an important role in the solution of parabolic equations where A results from some form of discretization of an elliptic o...

We give a simple condition on a matrix for which if the exponential matrix is diagonal, lower or upper triangular, then so is . It is also shown that for diagonalizable and any matrix , and commute if and only if and commute. These results are useful in problems in which knowledge about has to be extracted from structural information about its exponential, such as in large-scale sampled-data sy...

In this paper, we study the approximation of matrix-exponential distributions by Coxian distributions. Based on the spectral polynomial algorithm, we develop an algorithm for computing Coxian representations of Coxian distributions that are approximations of matrix-exponential distributions. As a specialization, we show that phase-type (PH) distributions can be approximated by Coxian distributi...

Many different algorithms have been suggested for computing the matrix exponential. In this paper, we put forward the idea of expanding in either Chebyshev, Legendre or Laguerre orthogonal polynomials. In order for these expansions to converge quickly, we cluster the eigenvalues into diagonal blocks and accelerate using shifting and scaling.

An efficient numerical method is developed for evaluating φ(A), where A is a symmetric matrix and φ is the function defined by φ(x) = (ex − 1)/x = 1+ x/2 + x2/6+ .... This matrix function is useful in the so-called exponential integrators for differential equations. In particular, it is related to the exact solution of the ODE system dy/dt = Ay + b, where A and b are t-independent. Our method a...

Based on the general concept of randomization, we develop linear-algebraic approximations for continuous probability distributions that involve the exponential of a matrix in their definitions, such as phase types and matrixexponential distributions. The approximations themselves result in proper probability distributions. For such a global randomization with the Erlangdistribution, we show tha...

We discuss a method to obtain closed-form expressions of f(A), where f is an analytic function and A a square, diagonalizable matrix. The method exploits the Cayley–Hamilton theorem and has been previously reported using tools that are perhaps not sufficiently appealing to physicists. Here, we derive the results on which the method is based by using tools most commonly employed by physicists. W...