Takagi’s decomposition of a symmetric unitary matrix as a finite algorithm

Takagi’s decomposition is an analog (for complex symmetric matrices and for unitary similarities replaced by unitary congruences) of the eigenvalue decomposition of Hermitian matrices. It is shown that, if a complex matrix is not only symmetric but is also unitary, then its Takagi decomposition can be found by quadratic radicals, that is, by means of a finite algorithm that involves arithmetic operations and quadratic radicals. A similar fact is valid for the eigenvalue decomposition of reflections, which are Hermitian unitary matrices.     

Relative invariants of 3 × 3 matrix triples ∗

We present primary and secondary generators for the algebra of polynomial invariants of the direct product of two copies of the special linear group Sl ₃ acting naturally on triples of 3 × 3 matrices over a field of characteristic zero. We handle also the analogous problem for triples and quadruples of 2 × 2 matrices.     

Polar and singular value decomposition of 3×3 magic squares

In the literature there have been reported a number of studies of probabilities involved in Monopoly, the worldwide popular game designed by Charles B. Darrow in the 1930s. Simple assumptions often lead to inaccurate results. In this paper, having distinguished between ‘In Jail’ and ‘Just Visiting’, both theoretical computations and computer simulations are conducted to determine the probability of landing on ‘JAIL’. Theoretical and simulation results are consistent. The sum of the probabilities of landing on ‘In Jail’ and ‘Just Visiting’ is found to be 6.40%, both in theory and in simulation, compared with Ian Stewart's 5.89%. The long-term probabilities of landing on the other 39 spaces are computed and it is concluded that the best properties to buy are the orange ones.     

An investigation of feasible descent algorithms for?estimating the condition number of a matrix

Techniques for estimating the condition number of a nonsingular matrix are developed. It is shown that Hager??s 1-norm condition number estimator is equivalent to the conditional gradient algorithm applied to the problem of maximizing the 1-norm of a matrix-vector product over the unit sphere in the 1-norm. By changing the constraint in this optimization problem from the unit sphere to the unit simplex, a new formulation is obtained which is the basis for both conditional gradient and projected gradient algorithms. In the test problems, the spectral projected gradient algorithm yields condition number estimates at least as good as those obtained by the previous approach. Moreover, in some cases, the spectral gradient projection algorithm, with a careful choice of the parameters, yields improved condition number estimates.     

On the rank of 3×3×3-tensors

Let U, V and W be finite dimensional vector spaces over the same field. The rank of a tensor τ in U???V???W is the minimum dimension of a subspace of U???V???W containing τ and spanned by fundamental tensors, i.e. tensors of the form u???v???w for some u in U, v in V and w in W. We prove that if U, V and W have dimension three, then the rank of a tensor in U???V???W is at most six, and such a bound cannot be improved, in general. Moreover, we discuss how the techniques employed in the proof might be extended to prove upper bounds for the rank of a tensor in U???V???W when the dimensions of U, V and W are higher.     

Conversion of a rhotrix to a ‘coupled matrix’

In this note, a method of converting a rhotrix to a special form of matrix termed a ‘coupled matrix’ is proposed. The special matrix can be used to solve various problems involving n?×?n and (n?–?1)?×?(n?–?1) matrices simultaneously.     

An algorithm to compute <Emphasis Type="Italic">ω</Emphasis>-primality in a numerical monoid

Let M be a commutative, cancellative, atomic monoid and x a nonunit in M. We define ω(x)=n if n is the smallest positive integer with the property that whenever x∣a ₁???a _t , where each a _i is an atom, there is a T?{1,2,…,t} with |T|≤n such that x∣∏_k∈T a _k . The ω-function measures how far x is from being prime in M. In this paper, we give an algorithm for computing ω(x) in any numerical monoid. Simple formulas for ω(x) are given for numerical monoids of the form 〈n,n+1,…,2n?1〉, where n≥3, and 〈n,n+1,…,2n?2〉, where n≥4. The paper then focuses on the special case of 2-generator numerical monoids. We give a formula for computing ω(x) in this case and also necessary and sufficient conditions for determining when x is an atom. Finally, we analyze the asymptotic behavior of ω(x) by computing \(\lim_{x\rightarrow \infty}\frac{\omega(x)}{x}\).     

Some algebras similar to the 2×2 Jordanian matrix algebra

The impetus for this study is the work of Dumas and Rigal on the Jordanian deformation of the ring of coordinate functions on 2×2 matrices. We are also motivated by current interest in birational equivalence of noncommutative rings. Recognizing the construction of the Jordanian matrix algebra as a skew polynomial ring, we construct a family of algebras relative to differential operator rings over a polynomial ring in one variable which are birationally equivalent to the Weyl algebra over a polynomial ring in two variables.     

On a minimal set of generators for the invariants of 3 × 3 matrices

Let K be a field of characteristic p > 0, let L be a restricted Lie algebra and let R be an associative K-algebra. It is shown that the various constructions in the literature of crossed product of R with u(L) are equivalent. We calculate explicit formulae relating the parameters involved and obtain a formula which hints at a noncommutative version of the Bell polynomials.     

An Integer Linear Programming approach to minimize the cost of the refurbishment of a façade to improve the energy efficiency of a building

Buildings account 40% of the EU's total energy consumption. Therefore, they represent a key potential source of energy savings to fight, among others, against climate change. Furthermore, around 54% of the buildings in Spain date back before 1980, when no thermal regulation was available. The refurbishment of a façade of an old building is usually the most effective way to improve its energy efficiency, by adding layers to the external envelope in order to reduce its thermal transmittance. This paper deals with the problem of minimizing costs for the thermal refurbishment of a façade with thickness and thermal transmittance bounds and with an intervention both on the opaque part (wall) and the transparent part (windows). Among thousands, even millions of combinations of materials and thicknesses for the different layers to be added to the opaque part, types of frame, and combinations of glasses and air chambers for the transparent part, the aim is to choose the one that minimizes the cost without violating any restriction imposed to the thermal refurbishment, in particular the current energy efficiency regulations in the zone. To optimally solve this problem, it will be modelled as an Integer Linear Programming problem with binary variables. The case study will be Building 1B of the School for Building Engineering of the Polytechnic University of Valencia, Spain. It was built in the late 1960s and has had a very inefficient energy consumption record. The optimal solution will be found among more than 6 million feasible solutions.     

The role of the isotonizing algorithm in Stein’s covariance matrix estimator

Covariance matrix estimation is central to many applications in statistics and allied fields. A useful estimator in this context was proposed by Stein which regularizes the sample covariance matrix by shrinking its eigenvalues together. This estimator can sometimes yield estimates of the eigenvalues that are negative or differ in order from the observed eigenvalues. In order to rectify this problem, Stein also proposed an ad hoc “isotonizing” procedure which pools together eigenvalue estimates in such a way that the original ordering and positivity of the estimates are enforced. From numerical studies, Stein’s “isotonized” estimator is known to have good risk properties in comparison with the maximum likelihood estimator. However, it remains unclear what role is played by the isotonizing procedure in the remarkable risk reductions achieved by Stein’s estimator. Through two distinct lines of investigations, it is established that Stein’s estimator without the isotonizing algorithm gives only modest risk reductions. In cases where the isotonizing algorithm is frequently used, however, Stein’s estimator can lead to significant risk reductions for certain domains of the parameter. In other cases, Stein’s estimator can even yield risk increases, such as when (1) the theoretical eigenvalues are well separated, and/or (2) when the sample size is moderate to large, leading to over-shrinkage.     

The complex step approximation to the Fréchet derivative of a matrix function

We show that the Fréchet derivative of a matrix function f at A in the direction E, where A and E are real matrices, can be approximated by Im f(A + ihE)/h for some suitably small h. This approximation, requiring a single function evaluation at a complex argument, generalizes the complex step approximation known in the scalar case. The approximation is proved to be of second order in h for analytic functions f and also for the matrix sign function. It is shown that it does not suffer the inherent cancellation that limits the accuracy of finite difference approximations in floating point arithmetic. However, cancellation does nevertheless vitiate the approximation when the underlying method for evaluating f employs complex arithmetic. The ease of implementation of the approximation, and its superiority over finite differences, make it attractive when specialized methods for evaluating the Fréchet derivative are not available, and in particular for condition number estimation when used in conjunction with a block 1-norm estimation algorithm.