The generalized spectral decomposition expressing a linear operator as a sum of idempotent and nilpotent operators is derived from the partial fraction decomposition of the reciprocal of its minimal polynomial, and makes possible a functional calculus of analytic functions of the operator. The classical spectral and polar decompositions for matrices are shown to be easy consequences. The idempo...

In this paper, we present a new and efficient algorithm for computing minimal polynomial of matrices over algebraic extension fields using the Gröbner bases technique. We have implemented our algorithm in Maple and we evaluate its performance and compare it to the performance of the function MinimalPolynomial of Maple 15 and also of the Bia las algorithm as a new algorithm to compute minimal po...

Let O be an order of discriminant D in an imaginary quadratic field K. In Lecture 19 we saw that there is a one-to-one relationship between isomorphism classes of elliptic curves with complex multiplication by O (the set EllO(C)), and equivalence classes of proper Oideals (the group cl(O)). The first main theorem of complex multiplication states that the elements of EllO(C) are algebraic intege...

Let V be an n-dimensional vector space over the finite field Fq and T a linear operator on V . For each k ∈ {1, . . . , n} we determine the number of k-dimensional T -invariant subspaces of V . Finally, this method is applied for the enumeration of all monomially nonisometric linear (n, k)-codes over Fq . 0. Introduction Let q be a power of a prime p, Fq the finite field with q elements and n a...

We show that the number α = (1+ √ 3 + 2 √ 5)/2 with minimal polynomial x4−2x3+x−1 is the only Pisot number whose four distinct conjugates α1, α2, α3, α4 satisfy the additive relation α1 + α2 = α3 + α4. This implies that there exists no two non-real conjugates of a Pisot number with the same imaginary part and also that at most two conjugates of a Pisot number can have the same real part. On the...

This note can be seen as an appendix of a recent paper of Kim [Kim15]. We show that the discrete logarithm problem in fields FQ where Q = p with p of medium size and n having a factor of the good size (specified in the article) has a complexity of LQ(1/3, 3 √ 48/9). We propose here a variant of NFS which combines exTNFS with the Conjugation method of polynomial selection. Let Q = p where n = ηk...

A primitive matrix is a square matrix M with nonnegative real entries such that the entries of M are all positive for some positive integer s. The smallest such s is called the primitivity index of M . Primitive matrices of normal type (namely: MM and MM have the same zero entries) occur naturally in studying the so called ”conjugacy-class covering number”and ”character covering number”of a …ni...

An (f(x); m)-interleaved sequence Gong (IEE Trans. Inform. Theory 41(2) (1995) 400) over a !nite !eld Fq is a sequence where all its m-decimated sequences have f(x) as the common characteristic polynomial, which can be determined by a control parameter. This paper demonstrates how issues such as minimal polynomial, linear span, period and correlation values of an (f(x); m)-interleaved sequence ...

In this paper, we consider some aspects related to determining the linear complexity of sequences over GF(2n). In particular, we study the effect of changing the finite field basis on the minimal polynomials, and thus on the linear complexity, of sequences defined overGF(2n) but given in their binary representation. Let a={ai} be a sequence overGF(2n). Then ai can be represented by ai = ∑n−1 j=...

A fundamental part of a fault diagnosis system is the residual generator. Here a new method, the minimal polynomial basis approach, for design of residual generators for linear systems, is presented. The residual generation problem is transformed into a problem of finding polynomial bases for null-spaces of polynomial matrices. This is a standard problem in established linear systems theory, wh...