Principal Logarithm Research Articles

An open set of 4×4 embeddable matrices whose principal logarithm is not a Markov generator

ABSTRACT A Markov matrix is embeddable if it can represent a homogeneous continuous-time Markov process. It is well known that if a Markov matrix has real and pairwise-different eigenvalues, then the embeddability can be determined by checking whether its principal logarithm is a rate matrix or not. The same holds for Markov matrices that are close enough to the identity matrix. In this paper we exhibit open sets of Markov matrices that are embeddable and whose principal logarithm is not a rate matrix, thus proving that the principal logarithm test above does not suffice generically.

Quantum walks on embeddings

We introduce a new type of discrete quantum walks, called vertex-face walks, based on orientable embeddings. We first establish a spectral correspondence between the transition matrix U and the vertex-face incidence structure. Using the incidence graph, we derive a formula for the principal logarithm of \(U^2\), and find conditions for its underlying digraph to be an oriented graph. In particular, we show this happens if the vertex-face incidence structure forms a partial geometric design. We also explore properties of vertex-face walks on the covers of a graph. Finally, we study a non-classical behavior of vertex-face walks.

Frequency Truncated Discrete-Time System Norm

Computation of the frequency truncated norm of a discrete-time linear time-invariant system is required in multi-rate digital signal processing and model reduction applications. For a system represented using state-space equations, the frequency truncated norm itself is a particular case of an integral of the transfer function of a discrete-time linear time-invariant descriptor system over a limited frequency band. This letter provides closed-form expressions for this integral and the frequency truncated norm for direct computation in terms of the principal logarithm and matrix exponential function.

Verified computation for the matrix principal logarithm

Two iterative algorithms are proposed for numerically computing an interval matrix containing the matrix principal logarithm. The first algorithm is based on a numerical spectral decomposition and requires only cubic complexity per iteration. The second algorithm is based on a numerical Jordan decomposition and applicable even for defective matrices. The complexity per iteration of the second algorithm is quartic.

A Logarithmic Minimization Property of the Unitary Polar Factor in the Spectral and Frobenius Norms

The unitary polar factor $Q=U_p$ in the polar decomposition of $Z=U_p \, H$ is the minimizer over unitary matrices $Q$ for both $\|{\rm Log}(Q^* Z)\|^2$ and its Hermitian part $\|{{\rm sym}{_{_*}}\!}({\rm Log}(Q^* Z))\|^2$ over both $\mathbb{R}$ and $\mathbb{C}$ for any given invertible matrix $Z\in\mathbb{C}^{n\times n}$ and any matrix logarithm Log, not necessarily the principal logarithm log. We prove this for the spectral matrix norm for any $n$ and for the Frobenius matrix norm for $n\leq 3$. The result shows that the unitary polar factor is the nearest orthogonal matrix to $Z$ not only in the normwise sense but also in a geodesic distance. The derivation is based on Bhatia's generalization of Bernstein's trace inequality for the matrix exponential and a new sum of squared logarithms inequality. Our result generalizes the fact for scalars that for any complex logarithm and for all $z\in\mathbb{C}{\setminus}\{0\} \min_{\vartheta\in(-\pi,\pi]}{|{\rm Log}_{\mathbb{C}}(e^{-i\vartheta} z)|}^2={|{\rm log}|...

Effects of arterial strain and stress in the prediction of restenosis risk: Computer modeling of stent trials

In-stenting restenosis is one of the major complications after stenting. Clinical trials of various stent designs have reported different restenosis rates. However, quantitative correlation between stent features and restenosis statistics is scant. In this work, it is hypothesized that stress concentrations on arterial wall caused artery injury, which initiates restenosis. The goal is to assess the correlation between stent-induced arterial stress and strain and the documented restenosis rates Six commercially available stents, including balloon-expandable stents and self-expanding stents, were virtually implanted into the arteries through finite element method. The resulted peak Von Mises stress, principal stress, principal logarithm strain, as well as percentage of intimal area with abnormal higher stress were monitored. Positive correlation between arterial stress and strain after stent implantations and the documented restenosis rates from the corresponding clinical trials was found regardless of stent types. No statistical significant difference was observed for various stress or strain parameters serving as indicators of artery injury. In-stent restenosis are less likely to occur as arterial mechanics are least altered by stent implantations. Optimization of stent designs to minimize the stent-induced arterial stresses and strains can reduce the arterial injury, and thus reduce the occurrence of restenosis. This work improved our understanding of the stent-lesion interactions that regulate arterial mechanics and demonstrated that arterial stress and strain could predict the risk of in-stent restenosis.

An explicit formula for the matrix logarithm

No Abstract Available

Conditioning and Padé Approximation of the Logarithm of a Matrix

In this work we (i) use the theory of piecewise analytic functions to represent the Fréchet derivative of any primary matrix function, in particular of primary logarithms; (ii) propose an indicator to assess inherent difficulties to compute a logarithm; and (iii) revisit Padé approximation techniques for the principal logarithm of block triangular matrices.