scholarly journals The Exact Distribution of the Condition Number of Complex Random Matrices

2013 ◽  
Vol 2013 ◽  
pp. 1-4
Author(s):  
Lin Shi ◽  
Taibin Gan ◽  
Hong Zhu ◽  
Xianming Gu
Keyword(s):  
Random Matrices ◽  
Condition Number ◽  
Random Matrix ◽  
Singular Values ◽  
Exact Distribution ◽  
Wishart Matrix ◽  
Zonal Polynomials ◽  
Wishart Matrices

LetGm×n (m≥n)be a complex random matrix andW=Gm×nHGm×nwhich is the complex Wishart matrix. Letλ1>λ2>…>λn>0andσ1>σ2>…>σn>0denote the eigenvalues of theWand singular values ofGm×n, respectively. The 2-norm condition number ofGm×nisκ2Gm×n=λ1/λn=σ1/σn. In this paper, the exact distribution of the condition number of the complex Wishart matrices is derived. The distribution is expressed in terms of complex zonal polynomials.

scholarly journals Dynamical correlation functions for products of random matrices

2015 ◽  
Vol 04 (04) ◽  
pp. 1550020 ◽  
Author(s):  
Eugene Strahov
Keyword(s):  
Random Matrices ◽  
Discrete Time ◽  
Random Matrix ◽  
Correlation Functions ◽  
Singular Values ◽  
Matrix Product ◽  
Dynamical Correlation ◽  
Products Of Random Matrices ◽  
Special Cases ◽  
Correlation Kernels

We introduce and study a family of random processes with a discrete time related to products of random matrices. Such processes are formed by singular values of random matrix products, and the number of factors in a random matrix product plays a role of a discrete time. We consider in detail the case when the (squared) singular values of the initial random matrix form a polynomial ensemble, and the initial random matrix is multiplied by standard complex Gaussian matrices. In this case, we show that the random process is a discrete-time determinantal point process. For three special cases (the case when the initial random matrix is a standard complex Gaussian matrix, the case when it is a truncated unitary matrix, or the case when it is a standard complex Gaussian matrix with a source) we compute the dynamical correlation functions explicitly, and find the hard edge scaling limits of the correlation kernels. The proofs rely on the Eynard–Mehta theorem, and on contour integral representations for the correlation kernels suitable for an asymptotic analysis.

scholarly journals On the noncentral distribution of the ratio of the extreme roots of wishart matrix

1981 ◽  
Vol 4 (1) ◽  
pp. 147-154
Author(s):  
V. B. Waikar
Keyword(s):  
Nuclear Physics ◽  
Normal Population ◽  
Wishart Distribution ◽  
Exact Distribution ◽  
Multivariate Normal ◽  
Wishart Matrix ◽  
Zonal Polynomials ◽  
Characteristic Roots ◽  
Multivariate Normal Population ◽  
Mean Vector

The distribution of the ratio of the extreme latent roots of the Wishart matrix is useful in testing the sphericity hypothesis for a multivariate normal population. LetXbe ap×nmatrix whose columns are distributed independently as multivariate normal with zero mean vector and covariance matrix∑. Further, letS=XX′and let11>…>1p>0be the characteristic roots ofS. ThusShas a noncentral Wishart distribution. In this paper, the exact distribution offp=1−1p/11is derived. The density offpis given in terms of zonal polynomials. These results have applications in nuclear physics also.

scholarly journals On minimal singular values of random matrices with correlated entries

2015 ◽  
Vol 04 (02) ◽  
pp. 1550006 ◽  
Author(s):  
F. Götze ◽  
A. Naumov ◽  
A. Tikhomirov
Keyword(s):  
Random Matrices ◽  
Random Matrix ◽  
Singular Values ◽  
Singular Value ◽  
The Matrix ◽  
Class Of Matrices ◽  
Least Singular Value ◽  
Mutually Independent

Let X be a random matrix whose pairs of entries Xjk and Xkj are correlated and vectors (Xjk, Xkj), for 1 ≤ j < k ≤ n, are mutually independent. Assume that the diagonal entries are independent from off-diagonal entries as well. We assume that [Formula: see text], for any j, k = 1, …, n and 𝔼 XjkXkj = ρ for 1 ≤ j < k ≤ n. Let Mn be a non-random n × n matrix with ‖Mn‖ ≤ KnQ, for some positive constants K > 0 and Q ≥ 0. Let sn(X + Mn) denote the least singular value of the matrix X + Mn. It is shown that there exist positive constants A and B depending on K, Q, ρ only such that [Formula: see text] As an application of this result we prove the elliptic law for this class of matrices with non-identically distributed correlated entries.

scholarly journals Singular values and evenness symmetry in random matrix theory

2016 ◽  
Vol 28 (5) ◽  
pp. 873-891 ◽  
Author(s):  
Folkmar Bornemann ◽  
Peter J. Forrester
Keyword(s):  
Random Matrices ◽  
Random Matrix ◽  
Functional Form ◽  
Matrix Theory ◽  
Stereographic Projection ◽  
Singular Values ◽  
Unitary Symmetry ◽  
Matrix Ensembles ◽  
Gap Probabilities ◽  
Orthogonal Ensembles

AbstractComplex Hermitian random matrices with a unitary symmetry can be distinguished by a weight function. When this is even, it is a known result that the distribution of the singular values can be decomposed as the superposition of two independent eigenvalue sequences distributed according to particular matrix ensembles with chiral unitary symmetry. We give decompositions of the distribution of singular values, and the decimation of the singular values – whereby only even, or odd, labels are observed – for real symmetric random matrices with an orthogonal symmetry, and even weight. This requires further specifying the functional form of the weight to one of three types – Gauss, symmetric Jacobi or Cauchy. Inter-relations between gap probabilities with orthogonal and unitary symmetry follow as a corollary. The Gauss case has appeared in a recent work of Bornemann and La Croix. The Cauchy case, when appropriately specialised and upon stereographic projection, gives decompositions for the analogue of the singular values for the circular unitary and circular orthogonal ensembles.

VICTORIA transform, RESPECT and REFORM methods for the proof of the G-Elliptic Law under G-Lindeberg condition and twice stochastic condition for the variances and covariances of the entries of some random matrices

2020 ◽  
Vol 28 (2) ◽  
pp. 131-162
Author(s):  
Vyacheslav L. Girko
Keyword(s):  
Random Matrices ◽  
Random Matrix ◽  
Lindeberg Condition

AbstractThe G-Elliptic law under the G-Lindeberg condition for the independent pairs of the entries of a random matrix is proven.

scholarly journals The Expected Characteristic and Permanental Polynomials of the Random Gram Matrix

10.37236/3709 ◽  
2014 ◽  
Vol 21 (3) ◽  
Author(s):  
Jacob G. Martin ◽  
E. Rodney Canfield
Keyword(s):  
Random Matrix ◽  
Generating Functions ◽  
Numerical Algorithms ◽  
Singular Values ◽  
Gram Matrix ◽  
Efficient Computation ◽  
Underlying Distribution ◽  
Characteristic Roots ◽  
Speed Up ◽  
Theoretical Results

A $t \times n$ random matrix $A$ can be formed by sampling $n$ independent random column vectors, each containing $t$ components. The random Gram matrix of size $n$, $G_{n}=A^{T}A$, contains the dot products between all pairs of column vectors in the randomly generated matrix $A$, and has characteristic roots coinciding with the singular values of $A$. Furthermore, the sequences $\det{(G_{i})}$ and $\text{perm}(G_{i})$ (for $i = 0, 1, \dots, n$) are factors that comprise the expected coefficients of the characteristic and permanental polynomials of $G_{n}$. We prove theorems that relate the generating functions and recursions for the traces of matrix powers, expected characteristic coefficients, expected determinants $E(\det{(G_{n})})$, and expected permanents $E(\text{perm}(G_{n}))$ in terms of each other. Using the derived recursions, we exhibit the efficient computation of the expected determinant and expected permanent of a random Gram matrix $G_{n}$, formed according to any underlying distribution. These theoretical results may be used both to speed up numerical algorithms and to investigate the numerical properties of the expected characteristic and permanental coefficients of any matrix comprised of independently sampled columns.

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