Boolean functions used in stream ciphers should have many cryptographic properties in order to help resist different kinds of cryptanalytic attacks. The resistance of Boolean functions against fast algebraic attacks is an important cryptographic property. Deciding the resistance of an n-variable Boolean function against fast algebraic attacks needs to determine the rank of a square matrix of or...

The article is devoted to different aspects of the question: ”What can be done with a complex-valued matrix by a low rank perturbation?” From the works of Thompson [15] we know how the Jordan normal form can be changed by a rank k perturbation, see Theorem 2. Particulary, it follows that one can do everything with a geometrically simple spectrum by a rank 1 perturbation, see Corollary 1. But th...

Abstract. The maximum likelihood estimation problem for matrices with bounded rank was studied in [1]. The study used numerical algebraic geometry to find the number of critical points of the likelihood function on the algebraic closure Vr of all complex matrices with rank less than or equal to r. In this paper we apply the EM algorithm to solve the maximum likelihood estimation problem for the...

The minimum rank of a simple graph G over a field F is the smallest possible rank among all symmetric matrices over F whose (i, j)th entry (for i = j) is nonzero whenever {i, j} is an edge in G and is zero otherwise. A universally optimal matrix is defined to be an integer matrix A such that every off-diagonal entry of A is 0, 1, or −1, and for all fields F , the rank of A is the minimum rank o...

We design two deterministic polynomial time algorithms for variants of a problem introduced by Edmonds in 1967: determine the rank of a matrix M whose entries are homogeneous linear polynomials over the integers. Given a linear subspace B of the n×n matrices over some field F, we consider the following problems: symbolic matrix rank (SMR) is the problem to determine the maximum rank among matri...

It is known that any square matrix A of size n over a field of prime characteristic p that has rank less than n/(p− 1) has a permanent that is zero. We give a new proof involving the invariant Xp. There are always matrices of any larger rank with non-zero permanents. It is shown that when the rank of A is exactly n/(p − 1), its permanent may be factorized into two functions involving Xp. Let n ...

Tian and Styan have shown many rank equalities for the sum of two and three idempotent matrices and pointed out that rank equalities for the sum P₁ + ⋯+P k with P₁,…, P k be idempotent (k > 3) are still open. In this paper, by using block Gaussian elimination, we obtained rank equalities for the sum of finitely many idempotent matrices and then solved the open problem mentioned above. Extension...

The paper mainly discusses the lower bounds for the rank of matrices and sufficient conditions for nonsingular matrices. We first present a new estimation for [Formula: see text] ([Formula: see text] is an eigenvalue of a matrix) by using the partitioned matrices. By using this estimation and inequality theory, the new and more accurate estimations for the lower bounds for the rank are deduced....

A natural family of quantized matrix algebras is introduced. It includes the two best studied such. Located inside Uq(A2n−1), it consists of quadratic algebras with the same Hilbert series as polynomials in n variables. We discuss their general properties and investigate some members of the family in great detail with respect to associated varieties, degrees, centers, and symplectic leaves. Fin...

Following the results of cite{Med}, regarding the Aluthge transform of polynomial matrices, the symbolic computation of the Duggal transform of a polynomial matrix $A$ is developed in this paper, using the polar decomposition and the singular value decomposition of $A$. Thereat, the polynomial singular value decomposition method is utilized, which is an iterative algorithm with numerical charac...