These bounds improved the corresponding results of cheng et al. The spectral radius of a finite graph is defined to be the spectral radius of its adjacency matrix. Moreover, there will always be a jsr achieving matrix product in which no matrix appears more than once. Some properties of the spectral radius of a set of matrices 185 3. Bounds on the spectral radius of a hadamard product of. Minimax theorem for the spectral radius of the product of non. When is the spectral radius of the product of two real matrices rhoab smaller or equal than the product of the spectral radii rhoarhob i have two real. Some new estimations for the upper and lower bounds for the. An estimation of the spectral radius of a product of block matrices meiqin chen a. Dec 25, 2016 the spectral radius of a square matrix is the largest of the moduli of all of its eigenvalues. Spectral radius, numerical radius, and the product of. It was shown that, for families of nonnegative matrices with independent column uncertainty, the maximal spectral radius is actually equal to the joint spectral radius. Motivated by and 14, 9, 10, in this paper we propose some inequalities on the upper bounds for the spectral radius of the hadamard product of any k nonnegative matrices. Hadamard product, nonnegative matrix, positive semide.
We present an elementary proof that the spectral radius of a matrix a may be obtained using the formula. Matrix norms and spectral radii after a reminder on norms and inner products, this lecture introduces the notions of matrix norm and induced matrix norm. Mathematics stack exchange is a question and answer site for people studying math at any level and professionals in related fields. Here x, y is the unitary inner product of the vectors x and y, and 1x1 x, x. The numerical range w a is a bounded convex set in c, and the numerical radius r a is a norm on b h satisfying. Studentnumericalanalysis spectralradius compute the spectral radius of a square matrix calling sequence parameters description notes examples calling sequence spectralradius a parameters a matrix. Spectral radius of matrix learn more about eigenvalues, spectral radius matlab. Then the relation between matrix norms and spectral radii is studied, culminating with gelfands formula for the spectral radius. Maximizing the spectral radius of a matrix product sciencedirect.
Joint spectral radius of rank one matrices and the maximum. Confusion between spectral radius of matrix and spectral radius of the operator. Spectral radius, hadamard product, kronecker product 1. It has an impact on the convergence of matrix power sequences and series. Introduction the spectral radius function is one of the most important functions of matrices. When initializing the weight matrix lets assume there is only one in an rnn recurrent neural network it is said e. It is closely related to matrix norms and the numerical radius. Spectral radius of nonnegative matrices 185 corollary 1. If anxn 2 0, r ra 0, then 0 spectral radius of a finite graph is defined to be the spectral radius of its adjacency matrix. In mathematics, the spectral radius of a square matrix or a bounded linear operator is the. In this paper, some new upper bounds on the spectral radius of the hadamard product of two nonnegative matrices and some upper and lower bounds on the spectral radius of the iterative matrix of a nonsingular mmatrix are given. In particular, the spectral radius of a nonnegative matrix is both.
As is known, the spectral radius of a matrix has a number of convexlike properties, see, e. I have a matrix defined such that each row has a sum lesser than one. A radius of a matrix a may be obtained using the formula. We prove the minimax equality for the spectral radius. On the other hand, if i read your question correctly, then you could replace rhoarhob by 1, as the spectral radius of a nonzero idempotent matrix is one. Introduction it is a wellknown fact from the theory of banach algebras that the. The proof of parts 1, 2 and 3 of the lemma can be found in golub and loan, 1996. It is a neat inequality and bears some symmetry, i.
A special case of such families was considered in the recent paper 1. Assuming for a moment that both matrices are diagonalisable which is always the case for the idempotent matrix. We prove zhans conjecture, and a related inequality for positive semidefinite matrices, using standard facts about principal submatrices, kronecker products, and the. When is the spectral radius of the product of two real. The purpose of this note is to prove zhans conjecture, and a little more. Some bounds for the spectral radius of the hadamard product. We form the matrices a i,b i as before and note that. Relation between the subordinate norm and the spectral radius of a matrix. Minimax theorem for the spectral radius of the product of nonnegative matrices. Definition of spectral radius eigenvalues of product of a.
A note on the spectral radius of a product of companion matrices. Spectral radius of the product of a right stochastic matrix and a block diagonal matrix. Assume the product of the m rectangular matrices is an nbyn square matrix. The numerical range and numerical radius are useful concepts in studying linear operators. The spectral radius of the iteration matrix is decisive for the convergence and stability, and the smaller it is, the faster the iteration method converges when the spectral radius is smaller than 1. Some inequalities on the spectral radius of matrices springerlink. Definition of spectral radius eigenvalues of product of a matrix and its complex conjugate transpose. Some inequalities on the spectral radius of matrices. Zhan has conjectured that the spectral radius of the hadamard product of two square nonnegative matrices is not greater than the spectral radius of their ordinary product. Citeseerx optimization of the spectral radius of a product.
In this paper, some upper bounds for the spectral radius. Let ra max iax, x l 1x1 1 be the numerical radius of a, and lla ll max 1. For notational inconvenience, we usually drop the matrix and regard the inner product as a scalar, i. Derivatives of the spectral radius as a function of non. In particular, the spectral radius of a nonnegative matrix is both quasiconvex and quasiconcave with respect to every row of a matrix as well as to its diagonal elements but not with respect to the whole matrix. The maximum absolute value of the n eigenvalues of the product matrix is called spectral radius. Spectral radius eigenvalues of product of a matrix and its complex conjugate transpose. There always exists an optimal product where no matrix appears more than once. These inequalities can improve considerably some previous results. Thanks for contributing an answer to mathematics stack exchange.
Minimax theorem for the spectral radius of the product of. Some inequalities on the spectral radius of matrices ncbi. In this paper, we study the limiting spectral radii of the product when m changes with n and can even diverge. Spectral radius of a matrix multiplied by a contraction matrix. Optimizing the spectral radius optimization online. A square matrix a is called diagonally symmetrizable for brevity, symmetrizable to a symmetric matrix s if it can be represented as a product a d lsd r, where d l and d r are positive diagonal matrices. Matrix derivatives math notation consider two vectors xand ywith the same number of components. Spectral radius of a product 231 similarly, the ratio to be estimated is homogeneous of degree zero in each of the matrices, so that both the ratio and the estimates are unchanged if one of the matrices is replaced by a positive multiple of itself. An estimation of the spectral radius of a product of block matrices. Citeseerx document details isaac councill, lee giles, pradeep teregowda. The spectral radius of a product of nonnegative matrices charles r. This in particular implies that the niteness property 1 always holds for rank one matrices.
Matrix inequalities for the fan product and the hadamard. Finiteness property and the optimal product corollary. Gelfands formula leads directly to a bound on the spectral radius of a product of finitely many matrices, namely assuming that they all commute we. A note on the spectral radius of a product of companion matrices 881 proof. Many functionals in matrix analysis are submultiplicative with respect to ordinary matrix multiplication, but the spectral. The spectral radius of a matrix is the maximum of the modulus of its eigenvalues. Some new estimations for the upper and lower bounds for. Spectral radius of the product of two matrices mathematics stack. This definition extends to the case of infinite graphs with bounded degrees of vertices i. However, as others have noticed, your whole code seems pretty mixed up and not actually valid matlab code, so your problem is not really to compute the spectral radius, is it. In this paper, some new upper bounds on the spectral radius of the hadamard product of two nonnegative matrices and some upper and lower bounds on the spectral radius of the iterative matrix of a nonsingular m matrix are given.
Zhan has conjectured 4 that for pairs of square nonnegative matrices, the spectral radius of the hadamard product is always bounded above by the spectral radius of the conventional matrix product. In addition, a new inequality on the spectral radius of the hadamard product of nonnegative matrices is also obtained. When is the spectral radius of the product of two real matrices rho. B1b2b k 1 since b1b2b k is a row stochastic matrix. An estimation of the spectral radius of a product of block. So far, many comparison theorems of single splittings of matrices have been presented in some papers and books 1 8. For simplicity, a product of two matrices is considered in the following theorem. I want to know if there is a theorem or result saying that the spectral radius of these kind of matrices is always lesser than. The jsr is achieved by the spectral radius of a finite matrix product, of length at most m. When a is nonnegative and irreducible, all the preceding weak inequalities 2 concerning the derivatives are replaced by strict inequalities. Let a be an nsquare complex matrix with eigenvalues 4. Definition of spectral radius eigenvalues of product of. Spectral radius of the product of a right stochastic matrix.
Sa attains a minimum and a maximum at a permutation matrix. On the ordering of spectral radius product a ad versus a d. For a nonnegative matrix a the spectral radius of the product xa is maximized over all nonnegative diagonal matrices x with trace 1. These bounds generalize some existing results, and some comparisons between these bounds are also considered. A new inequality on the minimum eigenvalue for the fan product of nonsingular mmatrices is given.
1543 418 1084 1322 1065 1151 577 853 1275 839 231 366 894 883 302 1437 1017 281 1099 508 1234 1010 1346 345 446 877 1279 512 967 315 152 146 265 1448 941 55 1012 291 424 1418 418 605 1154 856 1256