We establish a central limit theorem for the logarithm of the characteristic polynomial of a random permutation matrix. With this result we can obtain a central limit theorem for the counting function for the eigenvalues lying in some interval on the unit circle.

In [F. Uhlig, Explicit polar decomposition and a near-characteristic polynomial: The 2 × 2 case, Linear Algebra Appl., 38:239–249, 1981], the author gives a representation for the factors of the polar decomposition of a nonsingular real square matrix of order 2. Uhlig’s formulae are generalized to encompass all nonzero complex matrices of order 2 as well as all order n complex matrices with ran...

We consider the asymptotics of the second-order correlation function of the characteristic polynomial of a random matrix. We show that the known result for a random matrix from the Gaussian Unitary Ensemble essentially continues to hold for a general Hermitian Wigner matrix. Our proofs rely on an explicit formula for the exponential generating function of the second-order correlation function o...

Let M be an n × n square matrix and let p(λ) be a monic polynomial of degree n. Let Z be a set of n × n matrices. The multiplicative inverse eigenvalue problem asks for the construction of a matrix Z ∈ Z such that the product matrix MZ has characteristic polynomial p(λ). In this paper we provide new necessary and sufficient conditions when Z is an affine variety over an algebraically closed field.

Amitsur's formula, which expresses det(A + B) as a polynomial in coefficients of the characteristic polynomial of a matrix, is generalized for partial linearizations of the pfaffian of block matrices. As an applications, in up-coming papers we determine generators for the SO(n)-invariants of several matrices and relations for the O(n)-invariants of several matrices over a field of arbitrary cha...

We calculate joint moments of the characteristic polynomial of a random unitary matrix from the circular unitary ensemble and its derivative in the case that the power in the moments is an odd positive integer. The calculations are carried out for finite matrix size and in the limit as the size of the matrices goes to infinity. The latter asymptotic calculation allows us to prove a long-standin...

Let Aθ be the rotation C*-algebra for angle θ. For θ = p/q with p and q relatively prime, Aθ is the sub-C*-algebra of Mq(C(T )) generated by a pair of unitaries u and v satisfying uv = e2πiθvu. Let hθ,λ = u+u +λ/2(v+ v) be the almost Mathieu operator. By proving an identity of rational functions we show that for q even, the constant term in the characteristic polynomial of hθ,λ is (−1) q/2(1 + ...

For a prime p, we consider some natural classes of matrices over a finite field Fp of p elements, such as matrices of given rank or with characteristic polynomial having irreducible divisors of prescribed degrees. We demonstrate two different techniques which allow us to show that the number of such matrices in each of these classes and also with components in a given subinterval [−H,H] ⊆ [−(p ...

Let T be a self-affine tile in 2n defined by an integral expanding matrix A and a digit set D. The paper gives a necessary and sufficient condition for the connectedness of T. The condition can be checked algebraically via the characteristic polynomial of A. Through the use of this, it is shown that in 2#, for any integral expanding matrix A, there exists a digit set D such that the correspondi...

In this paper, we introduce weighted graph bundles and study their characteristic polynomial. In particular, we show that the characteristic polynomial of a weighted K2 (K2)bundles over a weighted graph F can be expressed as a product of characteristic polynomials two weighted graphs whose underlying graphs are F As an application, we compute the signature of a link whose corresponding weighted...