scholarly journals Computing The Permanent of (Some) Complex Matrices

2021 ◽  
pp. 753-761
Author(s):  
Pooja Mishra
Keyword(s):  
Relative Error ◽  
Deterministic Algorithm ◽  
Complex Matrix ◽  
Complex Matrices

We present a deterministic algorithm, which, for any given 0 < s <1 and an n × n real or complex matrix A = (aij) such that |aij − 1| ≤ 0.19 for all i, j computes the permanent of A within relative error s in nO(lnn−lns) time. The method can be extended to computing hafnians and multidimensional permanents.

Eigenvalue localization for complex matrices

2014 ◽  
Vol 27 ◽  
Author(s):  
Ibrahim Gumus ◽  
Omar Hirzallah ◽  
Fuad Kittaneh
Keyword(s):  
Frobenius Norm ◽  
Complex Matrix ◽  
Complex Matrices ◽  
Eigenvalue Localization ◽  
The Matrix

Let $A$ be an $n\times n$ complex matrix with $n\geq 3$. It is shown that at least $n-2$ of the eigenvalues of $A$ lie in the disk \begin{equation*}\left\vert z-\frac{\func{tr}A}{n}\right\vert \leq \sqrt{\frac{n-1}{n}\left(\sqrt{\left( \left\Vert A\right\Vert _{2}^{2}-\frac{\left\vert \func{tr} A\right\vert ^{2}}{n}\right) ^{2}-\frac{\left\Vert A^{\ast }A-AA^{\ast}\right\Vert _{2}^{2}}{2}}-\frac{\limfunc{spd}\nolimits^{2}(A)}{2}\right) },\end{equation*} where $\left\Vert A\right\Vert _{2},$ $\func{tr}A$, and $\limfunc{spd}(A)$ denote the Frobenius norm, the trace, and the spread of $A$, respectively. In particular, if $A=\left[ a_{ij}\right] $ is normal, then at least $n-2$ of the eigenvalues of $A$ lie in the disk {\small \begin{eqnarray*} & & \left\vert z-\frac{\func{tr}A}{n}\right\vert \\ & & \leq \sqrt{\frac{n-1}{n}\left( \frac{\left\Vert A\right\Vert _{2}^{2}}{2}-\frac{\left\vert \func{tr}A\right\vert ^{2}}{n}-\frac{3}{2}\max_{i,j=1,\dots,n} \left( \sum_{\substack{ k=1 \\ k\neq i}}^{n}\left\vert a_{ki}\right\vert ^{2}+\sum_{\substack{ k=1 \\ k\neq j}}^{n}\left\vert a_{kj}\right\vert ^{2}+\frac{\left\vert a_{ii}-a_{jj}\right\vert ^{2}}{2}\right) \right) }. \end{eqnarray*}} Moreover, the constant $\frac{3}{2}$ can be replaced by $4$ if the matrix $A$ is Hermitian.

A Unitary Relation Between a Matrix and its Transpose

1970 ◽  
Vol 13 (2) ◽  
pp. 279-280 ◽  
Author(s):  
William R. Gordon
Keyword(s):  
Simple Relation ◽  
Complex Matrix ◽  
Complex Matrices

It is well known that if A is an n × n complex matrix and AT is its transpose, then there is an invertible n x n complex matrix S such that AT = S-1AS. In this note we wish to point out another simple relation between A and AT.If A is an n × n complex matrix and AT is its transpose then there are unitary n × n complex matrices U and V such that AT = UAV.

Appendix: Real and complex canonical forms

2014 ◽  
Author(s):  
Leiba Rodman
Keyword(s):  
Canonical Forms ◽  
Real Matrix ◽  
Complex Matrix ◽  
Complex Matrices ◽  
Matrix Pencils ◽  
Real Matrices

This chapter presents canonical forms for real and complex matrices and for pairs of real and complex matrices, or matrix pencils, with symmetries. All these forms are known, and most are well-known. The chapter first looks at Jordan and Kronecker canonical forms, before turning to real matrix pencils with symmetries. It provides canonical forms for pairs of real matrices, either one of which is symmetric or skewsymmetric, or what is the same, corresponding matrix pencils. Finally, this chapter presents canonical forms of complex matrix pencils with various symmetries, such as complex matrix pencils with symmetries with respect to transposition.

scholarly journals On the Growth Order and Growth Type of Entire Functions of Several Complex Matrices

2020 ◽  
Vol 2020 ◽  
pp. 1-9 ◽  
Author(s):  
M. Abul-Ez ◽  
H. Abd-Elmageed ◽  
M. Hidan ◽  
M. Abdalla
Keyword(s):  
Entire Functions ◽  
Maximum Modulus ◽  
Growth Order ◽  
Complex Matrix ◽  
Growth Type ◽  
Complex Matrices ◽  
Taylor Coefficients ◽  
Linear Substitution ◽  
Higher Dimensional ◽  
Explicit Relation

In this paper, we establish an explicit relation between the growth of the maximum modulus and the Taylor coefficients of entire functions in several complex matrix variables (FSCMVs) in hyperspherical regions. The obtained formulas enable us to compute the growth order and the growth type of some higher dimensional generalizations of the exponential, trigonometric, and some special FSCMVs which are analytic in some extended hyperspherical domains. Furthermore, a result concerning linear substitution of the mode of increase of FSCMVs is given.

scholarly journals The determinant of a complex matrix and Gershgorin circles

2019 ◽  
Vol 35 ◽  
pp. 181-186 ◽  
Author(s):  
Florian Bünger ◽  
Siegfried Rump
Keyword(s):  
Linear Algebra ◽  
Connected Component ◽  
Complex Matrix ◽  
Outer Loop ◽  
Complex Matrices ◽  
Real Matrices

Each connected component of the Gershgorin circles of a matrix contains exactly as many eigenvalues as circles are involved. Thus, the Minkowski (set) product of all circles contains the determinant if all circles are disjoint. In [S.M. Rump. Bounds for the determinant by Gershgorin circles. Linear Algebra and its Applications, 563:215--219, 2019.], it was proved that statement to be true for real matrices whose circles need not to be disjoint. Moreover, it was asked whether the statement remains true for complex matrices. This note answers that in the affirmative. As a by-product, a parameterization of the outer loop of a Cartesian oval without case distinction is derived.

scholarly journals The pseudo core inverse of a companion matrix

2018 ◽  
Vol 55 (3) ◽  
pp. 407-420
Author(s):  
Yuefeng Gao ◽  
Jianlong Chen ◽  
Pedro Patrício ◽  
Dingguo Wang
Keyword(s):  
Toeplitz Matrix ◽  
Toeplitz Matrices ◽  
Generalized Inverses ◽  
Companion Matrix ◽  
Complex Matrix ◽  
Complex Matrices ◽  
The Core ◽  
Core Inverse ◽  
Companion Matrices ◽  
Existence Criteria

The notion of core inverse was introduced by Baksalary and Trenkler for a complex matrix of index 1. Recently, the notion of pseudo core inverse extended the notion of core inverse to an element of an arbitrary index in *-rings; meanwhile, it characterized the core-EP inverse introduced by Manjunatha Prasad and Mohana for complex matrices, in terms of three equations. Many works have been done on classical generalized inverses of companion matrices and Toeplitz matrices. In this paper, we discuss existence criteria and formulae of the pseudo core inverse of a companion matrix over a *-ring. A {1,3}-inverse of a Toeplitz matrix plays an important role in that process.

scholarly journals Algebraic Techniques for Least Squares Problems in Elliptic Complex Matrix Theory and Their Applications

2020 ◽  
Author(s):  
Hidayet Kösal ◽  
Müge Pekyaman
Keyword(s):  
Least Squares ◽  
Matrix Equation ◽  
Matrix Theory ◽  
Complex Matrix ◽  
Least Squares Problems ◽  
Real Numbers ◽  
Real Representation ◽  
Complex Matrices ◽  
Least Squares Solution ◽  
Elliptic Complex

In this study, we introduce concepts of norms of elliptic complex matrices and derive the least squares solution, the pure imaginary least squares solution, and the pure real least squares solution with the least norm for the elliptic complex matrix equation AX=B by using the real representation of elliptic complex matrices. To prove the authenticity of our results and to distinguish them from existing ones, some illustrative examples are also given. Elliptic numbers are generalized form of complex and so real numbers. Thus, the obtained results extend, generalize and complement some known least squares solutions results from the literature.

Application of pyrometer and standard sample to determine surface temperature of studied materials

2019 ◽  
pp. 9-13
Author(s):  
V.Ya. Mendeleyev ◽  
V.A. Petrov ◽  
A.V. Yashin ◽  
A.I. Vangonen ◽  
O.K. Taganov
Keyword(s):  
Composite Material ◽  
Surface Temperature ◽  
Relative Error ◽  
Wavelength Range ◽  
Standard Sample ◽  
A Priori ◽  
Temperature Changes ◽  
Surface Temperatures ◽  
Relative Errors ◽  
Normal Emissivity

Determining the surface temperature of materials with unknown emissivity is studied. A method for determining the surface temperature using a standard sample of average spectral normal emissivity in the wavelength range of 1,65–1,80 μm and an industrially produced Metis M322 pyrometer operating in the same wavelength range. The surface temperature of studied samples of the composite material and platinum was determined experimentally from the temperature of a standard sample located on the studied surfaces. The relative error in determining the surface temperature of the studied materials, introduced by the proposed method, was calculated taking into account the temperatures of the platinum and the composite material, determined from the temperature of the standard sample located on the studied surfaces, and from the temperature of the studied surfaces in the absence of the standard sample. The relative errors thus obtained did not exceed 1,7 % for the composite material and 0,5% for the platinum at surface temperatures of about 973 K. It was also found that: the inaccuracy of a priori data on the emissivity of the standard sample in the range (–0,01; 0,01) relative to the average emissivity increases the relative error in determining the temperature of the composite material by 0,68 %, and the installation of a standard sample on the studied materials leads to temperature changes on the periphery of the surface not exceeding 0,47 % for composite material and 0,05 % for platinum.

Seismite in the Devonian Sultan Formation, Frenchman Mountain, Nevada: Evidence of far-field effect of the Alamo Event

2016 ◽  
Vol 53 (2) ◽  
pp. 93-114
Author(s):  
Jesús Pinto ◽  
John Warme
Keyword(s):  
Carbonate Platform ◽  
Far Field ◽  
Carbonate Sediments ◽  
Impact Event ◽  
Complex Matrix ◽  
Near Surface ◽  
Fault Block ◽  
Southern Nevada ◽  
Salt Pan ◽  
The Alamo

We interpret a discrete, anomalous ~10-m-thick interval of the shallow-marine Middle to Late Devonian Valentine Member of the Sultan Formation at Frenchman Mountain, southern Nevada, to be a seismite, and that it was generated by the Alamo Impact Event. A suite of deformation structures characterize this unique interval of peritidal carbonate facies at the top of the Valentine Member; no other similar intervals have been discovered in the carbonate beds on Frenchman Mountain or in equivalent Devonian beds exposed in ranges of southern Nevada. The disrupted band extends for 5 km along the Mountain, and onto the adjoining Sunrise Mountain fault block for an additional 4+km. The interval displays a range of brittle, ductile and fluidized structures, and is divided into four informal bed-parallel units based on discrete deformation style and internal features that carry laterally across the study area. Their development is interpreted as the result of intrastratal compressional and contractional forces imposed upon the unconsolidated to fully cemented near-surface carbonate sediments at the top of the Valentine Member. The result is an assemblage of fractured, faulted, and brecciated beds, some of which were dilated, fluidized and injected to form new and complex matrix bands between beds. We interpret that the interval is an unusually thick and well displayed seismite. Because the Sultan Formation correlates northward to the Frasnian (lower Upper Devonian) carbonate rocks of the Guilmette Formation, and the Guilmette contains much thicker and more proximal exposures of the Alamo Impact Breccia, including seismites, we interpret the Frenchman Mountain seismite to be a far-field product of the Alamo Impact Event. Accompanying ground motion and deformation of the inner reaches of the Devonian carbonate platform may have resulted in a fall of relative sea level and abrupt shift to a salt-pan paleoenvironment exhibited by the post-event basal beds of the directly overlying Crystal Pass Member.

Sign in / Sign up

Export Citation Format

Share Document