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Square matrix

About: Square matrix is a research topic. Over the lifetime, 5000 publications have been published within this topic receiving 92428 citations.


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TL;DR: An analogue of the Daleckii--Krein theorem is state and proved, thus obtaining an explicit formula for the Frechet derivative of generalized matrix functions, and the differentiability of generalized Matrix functions of real matrices is proved under very mild assumptions.
Abstract: We state and prove an analogue of the Daleckii--Krein theorem, thus obtaining an explicit formula for the Frechet derivative of generalized matrix functions Moreover, we prove the differentiability of generalized matrix functions of real matrices under very mild assumptions For complex matrices, we argue that, under the same assumptions, generalized matrix functions are real-differentiable but generally not complex-differentiable Finally, we discuss the application of our results to the study of the condition number of generalized matrix functions Along our way, we also derive generalized matrix functional analogues of a few classical theorems on polynomial interpolation of classical matrix functions and their derivatives

24 citations

Journal ArticleDOI
TL;DR: In this article, the spectrum of an asymmetric random matrix with block structured variances is studied, where rows and columns of the random square matrix are divided into D partitions with arbitrary size and the parameters of the model are the variances of elements in each block.
Abstract: We study the spectrum of an asymmetric random matrix with block structured variances. The rows and columns of the random square matrix are divided into D partitions with arbitrary size (linear in N). The parameters of the model are the variances of elements in each block, summarized in g∈R+D×D. Using the Hermitization approach and by studying the matrix-valued Stieltjes transform we show that these matrices have a circularly symmetric spectrum, we give an explicit formula for their spectral radius and a set of implicit equations for the full density function. We discuss applications of this model to neural networks.

24 citations

01 Jan 1968
TL;DR: In this paper, necessary and sufficient conditions are obtained for a matrix A to have a g-inverse with rows and columns belonging to special linear manifolds, denoted by A - C.
Abstract: Necessary and sufficient conditions are obtained for a matrix A to have a g-inverse with rows and columns belonging to special linear manifolds. For a square matrix A, a g-inverse, with columns belonging to the linear manifold generated by the columns of A, is denoted by A - C . Such a g-inverse exists if and only if R(A)=R(A 2 ). The following properties of A - C are established: (a) A - C = A(A 2 ) - . (b) For any positive integer m, (A - C ) m provides a reflexive g-inverse of A m . (c) If x is an eigenvector corresponding to a nonnull eigenvalue λ of A, x is also an eigenvector of A - C corresponding to its eigenvalue 1/λ. The converse of this result is also true. (d) A special choice of (A 2 ) - =(A 3 )-A leads to A - C =A(A 3 )-A which is unique irrespective of the choice of (A 3 ) - and is, in fact, the same as the Scroggs-Odell pseudoinverse (J.SIAM 1966) of A. When R(A)=R(A 2 ), this indeed is a much simpler way of calculating the Scroggs-Odell pseudoinverse compared to the method indicated by its authors. (e) A(A 3 )-A belongs to the subalgebra generated by A.

24 citations

Journal ArticleDOI
TL;DR: A simple method is presented to expand the d matrix into a complex Fourier series and calculate the Fourier coefficients by exactly diagonalizing the angular momentum operator J(y) in the eigenbasis of J(z).
Abstract: The precise calculations of Wigner's d matrix are important in various research fields. Due to the presence of large numbers, direct calculations of the matrix using Wigner's formula suffer from a loss of precision. We present a simple method to avoid this problem by expanding the d matrix into a complex Fourier series and calculate the Fourier coefficients by exactly diagonalizing the angular momentum operator J(y) in the eigenbasis of J(z). This method allows us to compute the d matrix and its various derivatives for spins up to a few thousand. The precision of the d matrix from our method is about 10(-14) for spins up to 100.

24 citations

Journal ArticleDOI
TL;DR: In this paper, the theory of directed graphs is used to determine conditions under which the exponent is less than (n-1)2+1, and the best possible result is 2n-r-1.
Abstract: Let A be an n by n matrix with non-negative entries. If a permutation matrix P exists such that P-1 is of the form where A and C are square matrices and O is a zero-matrix then A is said to be reducible. Otherwise, A is irreducible. If A is irreducible, then A is said to be primitive if there is an integer k such that Ak0, i.e., Ak has no zero entries. If A is primitive, the least integer m for which Am0 is called the exponent of A. In (4), Wielandt has shown that for n by n primitive matrices the exponent is at most (n-1)2+1. In this paper, the theory of directed graphs is used to determine conditions under which the exponent is less than (n-1)2+1. The following results are obtained. If A contains r non-zero entries along its main diagonal then its exponent is at most 2n-r-1, and this result is the best possible. If all the diagonal entries of A are zero but its graph KA (see next section for definition) contains a cycle of length d, then the exponent of A is at most d(2n-d-1). For the case where this improves Wielandt's result.

24 citations


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Performance
Metrics
No. of papers in the topic in previous years
YearPapers
202322
202244
2021115
2020149
2019134
2018145