On the Matrix With Elements 1/(r+s-1)

1974 ◽  
Vol 17 (2) ◽  
pp. 297-298 ◽  
Author(s):  
W. W. Sawyer
Keyword(s):  
Tridiagonal Matrix ◽  
The Matrix ◽  
Eigenvectors And Eigenvalues

A standard example of a matrix for which the computation of eigenvectors and eigenvalues is very awkward is the matrix A with ars=1/(r+s—1), 1≤r≤n, 1 ≤s≤n. It is therefore of interest that A commutes with a tridiagonal matrix.

scholarly journals A note on the eigenvalue free intervals of some classes of signed threshold graphs

2019 ◽  
Vol 7 (1) ◽  
pp. 218-225
Author(s):  
Milica Anđelić ◽  
Tamara Koledin ◽  
Zoran Stanić
Keyword(s):  
Tridiagonal Matrix ◽  
Constant Sign ◽  
Equitable Partition ◽  
Elementary Matrix ◽  
Matrix Operations ◽  
Threshold Graphs ◽  
The Matrix ◽  
Threshold Graph

Abstract We consider a particular class of signed threshold graphs and their eigenvalues. If Ġ is such a threshold graph and Q(Ġ ) is a quotient matrix that arises from the equitable partition of Ġ , then we use a sequence of elementary matrix operations to prove that the matrix Q(Ġ ) – xI (x ∈ ℝ) is row equivalent to a tridiagonal matrix whose determinant is, under certain conditions, of the constant sign. In this way we determine certain intervals in which Ġ has no eigenvalues.

scholarly journals Bidiagonalization of (k, k + 1)-tridiagonal matrices

2019 ◽  
Vol 7 (1) ◽  
pp. 20-26 ◽  
Author(s):  
S. Takahira ◽  
T. Sogabe ◽  
T.S. Usuda
Keyword(s):  
Tridiagonal Matrix ◽  
Permutation Matrix ◽  
Block Diagonalization ◽  
Tridiagonal Matrices ◽  
The Matrix ◽  
Appl Math ◽  
Diagonalization Algorithm

Abstract In this paper,we present the bidiagonalization of n-by-n (k, k+1)-tridiagonal matriceswhen n < 2k. Moreover,we show that the determinant of an n-by-n (k, k+1)-tridiagonal matrix is the product of the diagonal elements and the eigenvalues of the matrix are the diagonal elements. This paper is related to the fast block diagonalization algorithm using the permutation matrix from [T. Sogabe and M. El-Mikkawy, Appl. Math. Comput., 218, (2011), 2740-2743] and [A. Ohashi, T. Sogabe, and T. S. Usuda, Int. J. Pure and App. Math., 106, (2016), 513-523].

scholarly journals On the determinantal structure of conditional overlaps for the complex Ginibre ensemble

2019 ◽  
Vol 09 (04) ◽  
pp. 2050015 ◽  
Author(s):  
Gernot Akemann ◽  
Roger Tribe ◽  
Athanasios Tsareas ◽  
Oleg Zaboronski
Keyword(s):  
Scaling Limits ◽  
Algebraic Decay ◽  
Ginibre Ensemble ◽  
Complex Eigenvalues ◽  
The Matrix ◽  
Number Formula ◽  
Overlap Matrix ◽  
The Right ◽  
Joint Evolution ◽  
Eigenvectors And Eigenvalues

We continue the study of joint statistics of eigenvectors and eigenvalues initiated in the seminal papers of Chalker and Mehlig. The principal object of our investigation is the expectation of the matrix of overlaps between the left and the right eigenvectors for the complex [Formula: see text] Ginibre ensemble, conditional on an arbitrary number [Formula: see text] of complex eigenvalues. These objects provide the simplest generalization of the expectations of the diagonal overlap ([Formula: see text]) and the off-diagonal overlap ([Formula: see text]) considered originally by Chalker and Mehlig. They also appear naturally in the problem of joint evolution of eigenvectors and eigenvalues for Brownian motions with values in complex matrices studied by the Krakow school. We find that these expectations possess a determinantal structure, where the relevant kernels can be expressed in terms of certain orthogonal polynomials in the complex plane. Moreover, the kernels admit a rather tractable expression for all [Formula: see text]. This result enables a fairly straightforward calculation of the conditional expectation of the overlap matrix in the local bulk and edge scaling limits as well as the proof of the exact algebraic decay and asymptotic factorization of these expectations in the bulk.

scholarly journals A Test Matrix for an Inverse Eigenvalue Problem

2014 ◽  
Vol 2014 ◽  
pp. 1-6 ◽  
Author(s):  
G. M. L. Gladwell ◽  
T. H. Jones ◽  
N. B. Willms
Keyword(s):  
Inverse Problem ◽  
Eigenvalue Problem ◽  
Explicit Solution ◽  
Additional Condition ◽  
Tridiagonal Matrix ◽  
Inverse Eigenvalue Problem ◽  
Test Matrix ◽  
Symmetric Tridiagonal Matrix ◽  
The Matrix ◽  
Inverse Eigenvalue

We present a real symmetric tridiagonal matrix of ordernwhose eigenvalues are{2k}k=0n-1which also satisfies the additional condition that its leading principle submatrix has a uniformly interlaced spectrum,{2l+1}l=0n-2. The matrix entries are explicit functions of the sizen, and so the matrix can be used as a test matrix for eigenproblems, both forward and inverse. An explicit solution of a spring-mass inverse problem incorporating the test matrix is provided.

scholarly journals The maximal \alpha-index of trees with k pendent vertices and its computation

2020 ◽  
Vol 36 (36) ◽  
pp. 38-46
Author(s):  
Oscar Rojo
Keyword(s):  
Spectral Radius ◽  
Adjacency Matrix ◽  
Diagonal Matrix ◽  
Tridiagonal Matrix ◽  
Tridiagonal Matrices ◽  
Symmetric Tridiagonal Matrix ◽  
The Matrix ◽  
Pendent Vertices ◽  
Equality Holding ◽  
Alpha Index

Let $G$ be a graph with adjacency matrix $A(G)$ and let $D(G)$ be the diagonal matrix of the degrees of $G$. The $\alpha-$ index of $G$ is the spectral radius $\rho_{\alpha}\left( G\right)$ of the matrix $A_{\alpha}\left( G\right)=\alpha D\left( G\right) +(1-\alpha)A\left( G\right)$ where $\alpha \in [0,1]$. Let $T_{n,k}$ be the tree of order $n$ and $k$ pendent vertices obtained from a star $K_{1,k}$ and $k$ pendent paths of almost equal lengths attached to different pendent vertices of $K_{1,k}$. It is shown that if $\alpha\in\left[ 0,1\right) $ and $T$ is a tree of order $n$ with $k$ pendent vertices then% \[ \rho_{\alpha}(T)\leq\rho_{\alpha}(T_{n,k}), \] with equality holding if and only if $T=T_{n,k}$. This result generalizes a theorem of Wu, Xiao and Hong \cite{WXH05} in which the result is proved for the adjacency matrix ($\alpha=0$). Let $q=[\frac{n-1}{k}]$ and $n-1=kq+r$, $0 \leq r \leq k-1$. It is also obtained that the spectrum of $A_{\alpha}(T_{n,k})$ is the union of the spectra of two special symmetric tridiagonal matrices of order $q$ and $q+1$ when $r=0$ or the union of the spectra of three special symmetric tridiagonal matrices of order $q$, $q+1$ and $2q+2$ when $r \neq 0$. Thus the $\alpha-$ index of $T_{n,k}$ can be computed as the largest eigenvalue of the special symmetric tridiagonal matrix of order $q+1$ if $r=0$ or order $2q+2$ if $r\neq 0$.

scholarly journals Motzkin Paths, Motzkin Polynomials and Recurrence Relations

10.37236/4781 ◽  
2015 ◽  
Vol 22 (2) ◽  
Author(s):  
Roy Oste ◽  
Joris Van der Jeugt
Keyword(s):  
Recurrence Relations ◽  
Catalan Numbers ◽  
Tridiagonal Matrix ◽  
Efficient Computation ◽  
Combinatorial Objects ◽  
Motzkin Numbers ◽  
The Matrix ◽  
Motzkin Paths

We consider the Motzkin paths which are simple combinatorial objects appearing in many contexts. They are counted by the Motzkin numbers, related to the well known Catalan numbers.  Associated with the Motzkin paths, we introduce the Motzkin polynomial, which is a multi-variable polynomial "counting" all Motzkin paths of a certain type. Motzkin polynomials (also called Jacobi-Rogers polynomials) have been studied before, but here we deduce some properties based on recurrence relations. The recurrence relations proved here also allow an efficient computation of the Motzkin polynomials. Finally, we show that the matrix entries of powers of an arbitrary tridiagonal matrix are essentially given by Motzkin polynomials, a property commonly known but usually stated without proof.

scholarly journals Matrices whose inverses are tridiagonal, band or block-tridiagonal and their relationship with the covariance matrices of a random Markov process

Filomat ◽  
2019 ◽  
Vol 33 (5) ◽  
pp. 1335-1352
Author(s):  
Ulan Brimkulov
Keyword(s):  
Markov Random Fields ◽  
Covariance Matrices ◽  
Tridiagonal Matrix ◽  
Matrix Elements ◽  
Banded Matrix ◽  
Markov Random ◽  
First Case ◽  
The Matrix ◽  
Full Dimension ◽  
Product Of Matrices

The article discusses the matrices of the form A1n, Amn, AmN, whose inverses are: tridiagonal matrix A-1n (n - dimension of the A-mn matrix), banded matrix A-mn (m is the half-width band of the matrix) or block-tridiagonal matrix A-m N (N = n x m - full dimension of the block matrix; m - the dimension of the blocks) and their relationships with the covariance matrices of measurements with ordinary (simple) Markov Random Processes (MRP), multiconnected MRP and vector MRP, respectively. Such covariance matrices frequently occur in the problems of optimal filtering, extrapolation and interpolation of MRP and Markov Random Fields (MRF). It is shown, that the structures of the matrices A1n, Amn, AmN have the same form, but the matrix elements in the first case are scalar quantities; in the second case matrix elements represent a product of vectors of dimension m; and in the third case, the off-diagonal elements are the product of matrices and vectors of dimension m. The properties of such matrices were investigated and a simple formulas of their inverses were found. Also computational efficiency in the storage and the inverse of such matrices have been considered. To illustrate the acquired results, an example on the covariance matrix inversions of two-dimensional MRP is given.

Crystalloid Inclusions in Hepatocyte Mitochondria and Their Similarity to Cytoplasmic Crystalloids

1974 ◽  
Vol 32 ◽  
pp. 198-199
Author(s):  
Odell T. Minick ◽  
Hidejiro Yokoo
Keyword(s):  
Liver Disease ◽  
Alcoholic Liver Disease ◽  
Special Interest ◽  
Structural Variations ◽  
Mitochondrial Alterations ◽  
The Matrix ◽  
Crystalloid Inclusions ◽  
Liver Biopsies

Mitochondrial alterations were studied in 25 liver biopsies from patients with alcoholic liver disease. Of special interest were the morphologic resemblance of certain fine structural variations in mitochondria and crystalloid inclusions. Four types of alterations within mitochondria were found that seemed to relate to cytoplasmic crystalloids.Type 1 alteration consisted of localized groups of cristae, usually oriented in the long direction of the organelle (Fig. 1A). In this plane they appeared serrated at the periphery with blind endings in the matrix. Other sections revealed a system of equally-spaced diagonal lines lengthwise in the mitochondrion with cristae protruding from both ends (Fig. 1B). Profiles of this inclusion were not unlike tangential cuts of a crystalloid structure frequently seen in enlarged mitochondria described below.

Coherency loss in γ' precipitates in nickel-base superalloy

1983 ◽  
Vol 41 ◽  
pp. 244-245
Author(s):  
R. A. Ricks ◽  
Angus J. Porter
Keyword(s):  
Lattice Mismatch ◽  
Lattice Parameter ◽  
Nickel Base Superalloy ◽  
Large Size ◽  
The Matrix ◽  
Nimonic 80A ◽  
Interfacial Dislocations ◽  
Nickel Base ◽  
Coherency Loss ◽  
Nimonic 115

During a recent investigation concerning the growth of γ' precipitates in nickel-base superalloys it was observed that the sign of the lattice mismatch between the coherent particles and the matrix (γ) was important in determining the ease with which matrix dislocations could be incorporated into the interface to relieve coherency strains. Thus alloys with a negative misfit (ie. the γ' lattice parameter was smaller than the matrix) could lose coherency easily and γ/γ' interfaces would exhibit regularly spaced networks of dislocations, as shown in figure 1 for the case of Nimonic 115 (misfit = -0.15%). In contrast, γ' particles in alloys with a positive misfit could grow to a large size and not show any such dislocation arrangements in the interface, thus indicating that coherency had not been lost. Figure 2 depicts a large γ' precipitate in Nimonic 80A (misfit = +0.32%) showing few interfacial dislocations.

Influence of Heat Treatments on Microstructures in an Fe-Co-V Alloy

1974 ◽  
Vol 32 ◽  
pp. 512-513
Author(s):  
S. Mahajan ◽  
M. R. Pinnel ◽  
J. E. Bennett
Keyword(s):  
Single Phase ◽  
Alloy Composition ◽  
Supersaturated Solid Solution ◽  
Cell Formation ◽  
Heat Treatments ◽  
Air Cooling ◽  
Iced Brine ◽  
Transmission Electron ◽  
The Matrix ◽  
Bcc Structure

The microstructural changes in an Fe-Co-V alloy (composition by wt.%: 2.97 V, 48.70 Co, 47.34 Fe and balance impurities, such as C, P and Ni) resulting from different heat treatments have been evaluated by optical metallography and transmission electron microscopy. Results indicate that, on air cooling or quenching into iced-brine from the high temperature single phase ϒ (fcc) field, vanadium can be retained in a supersaturated solid solution (α2) which has bcc structure. For the range of cooling rates employed, a portion of the material appears to undergo the γ-α2 transformation massively and the remainder martensitically. Figure 1 shows dislocation topology in a region that may have transformed martensitically. Dislocations are homogeneously distributed throughout the matrix, and there is no evidence for cell formation. The majority of the dislocations project along the projections of <111> vectors onto the (111) plane, implying that they are predominantly of screw character.

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