Narrow Results

We consider generalizations of commuting squares, called twisted commuting squares, obtained by having the commuting square orthogonality condition hold with respect to the inner product given by a faithful state on a finite-dimensional *-algebra. We present various examples of twisted commuting squares, most of which are computationally easy to work with, and we prove an isolation result. We also show how parametric families of (not necessarily) twisted commuting squares yield associative deformations of the matrix multiplication.

Algebraic theory of coding is one of modern fields of applications of algebra. This theory uses matrix algebra intensively. This paper is devoted to an application of Kronecker's matrix forms of presentations of the genetic code for algebraic analysis of a basic scheme of degeneracy of the genetic code. Similar matrix forms are utilized in the theory of signal processing and encoding. The Kronecker family of the genetic matrices is investigated, which is based on the genetic matrix [C A; U G], where C, A, U, G are the letters of the genetic alphabet. This matrix in the third Kronecker power is the (8*8)-matrix, which contains all 64 genetic triplets in a strict order with a natural binary numeration of the triplets by numbers from 0 to 63. Peculiarities of the basic scheme of degeneracy of the genetic code are reflected in the symmetrical black-and-white mosaic of this genetic (8*8)-matrix. This mosaic matrix is connected algorithmically with Hadamard matrices unexpectedly, which are famous in the theory of signal processing and encoding, spectral analysis, quantum mechanics and quantum computers. Furthermore, many kinds of cyclic permutations of genetic elements lead to reconstruction of initial Hadamard matrices into new Hadamard matrices unexpectedly. This demonstrates that matrix algebra is one of promising instruments and of adequate languages in bioinformatics and algebraic biology.

We characterize all connected graphs with exactly three distinct normalized Laplacian eigenvalues among which one is equal to 1, and determine all connected bipartite graphs with at least one vertex of degree 1 having exactly four distinct normalized Laplacian eigenvalues. In addition, we find all unicyclic graphs with three or four distinct normalized Laplacian eigenvalues.

We study the partial Hadamard matrices $H\in M_{M\times N}(ℂ)$ which are isolated, under the assumption that the entries $H_{ij}\in ℂ$ are roots of unity, or more generally, under the assumption that the combinatorics of H comes from vanishing sums of roots of unity. We first review the various conjectures on the subject, and then we present several new results, regarding notably the isolation of the master Hadamard matrices, $H_{ij}={\lambda }_i^{n_j}$, and the structure of the isolated matrices arising via the McNulty-Weigert construction. We discuss then the notion of isolation, in some related contexts, of the magic unitary matrices, and of the quantum permutation groups.

Consider the real space 𝔻_U of directions one can move in from a unitary N × N matrix U without disturbing its unitarity and the moduli of its entries in the first order. dim_ℝ (𝔻_U) is called the defect of U and denoted D(U). We give an account of Alexander Karabegov’s theory where 𝔻_U is parametrized by the imaginary subspace of the eigenspace, associated with λ = 1, of a certain unitary operator I_U on 𝕄_N, and where D(U) is the multiplicity of 1 in the spectrum of I_U. This characterization allows us to establish the dependence of D(U⁽¹⁾ ⊗ … ⊗U^(r)) on D(U^(k))’s, to derive formulas expressing D(F) for a Fourier matrix F of the size being a power of a prime, as well as to show the multiplicativity of D(F) with respect to Kronecker factors of F if their sizes are pairwise relatively prime. Also partly due to the role of symmetries of U in the determination of the eigenspaces of I_U we study the ‘permute and enphase’ symmetries and the equivalence of Fourier matrices, associated with arbitrary finite abelian groups. This work is divided in two parts — the present one and the second appearing in the next issue of OSID [1].

Consider the real space 𝔻_U of directions one can move in from a unitary N × N matrix U without disturbing its unitarity and the moduli of its entries in the first order. dim_ℝ (𝔻_U) is called the defect of U and denoted D(U). We give an account of Alexander Karabegov’s theory where 𝔻_U is parametrized by the imaginary subspace of the eigenspace, associated with λ = 1, of a certain unitary operator ℐ_U on 𝕄_N, and where D(U) is the multiplicity of 1 in the spectrum of ℐ_U. This characterisation allows us to establish the dependence of D(U⁽¹⁾ ⊗ … ⊗ U^(r)) on D(U^(k))’s, to derive formulas expressing D(F) for a Fourier matrix F of the size being a power of a prime number, as well as to show the multiplicativity of D(F) with respect to Kronecker factors of F if their sizes are pairwise relatively prime. Also partly due to the role of symmetries of U in the determination of the eigenspaces of ℐ_U we study the ‘permute and enphase’ symmetries and equivalence of Fourier matrices, associated with arbitrary finite abelian groups. This work is published as two papers — the first part [1] and the current second one.

For every Hadamard design with parameters $2$-$(n-1,\frac{n}{2}-1,\frac{n}{4}-1)$ having a skew-symmetric incidence matrix we give a construction of 54 Hadamard designs with parameters $2$-$(4n-1,2n-1,n-1)$. Moreover, for the case $n=8$ we construct doubly-even self-orthogonal binary linear codes from the corresponding Hadamard matrices of order 32. From these binary codes we construct five new extremal Type II ${\mathbb{Z}}_4$-codes of length 32. The constructed codes are the first examples of extremal Type II ${\mathbb{Z}}_4$-codes of length 32 and type $4^{k_1}{2}^{k_2}$, $k_1\in \{7,8,9,10\}$, whose residue codes have minimum weight 8. Further, correcting the results from the literature we construct 5147 extremal Type II ${\mathbb{Z}}_4$-codes of length 32 and type $4^{14}{2}^4$.