Common Invariant Subspace Research Articles

Constrained Common Invariant Subspace and Its Application

Constrained Common Invariant Subspace and Its Application

Algorithms for simultaneous block triangularization and block diagonalization of sets of matrices

<abstract><p>In a recent paper, a new method was proposed to find the common invariant subspaces of a set of matrices. This paper investigates the more general problem of putting a set of matrices into block triangular or block-diagonal form simultaneously. Based on common invariant subspaces, two algorithms for simultaneous block triangularization and block diagonalization of sets of matrices are presented. As an alternate approach for simultaneous block diagonalization of sets of matrices by an invertible matrix, a new algorithm is developed based on the generalized eigenvectors of a commuting matrix. Moreover, a new characterization for the simultaneous block diagonalization by an invertible matrix is provided. The algorithms are applied to concrete examples using the symbolic manipulation system Maple.</p></abstract>

A converse statement to Hutchinson’s theorem and a dimension gap for self-affine measures

A well-known theorem of J. E. Hutchinson states that if an iterated function system consists of similarity transformations and satisfies the open set condition then its attractor supports a self-similar measure with Hausdorff dimension equal to the similarity dimension. In this article we prove the following result which may be regarded as a form of partial converse: if an iterated function system consists of invertible affine transformations whose linear parts do not preserve a common invariant subspace, and its attractor supports a self-affine measure with Hausdorff dimension equal to the affinity dimension, then the system necessarily consists of similarity transformations. We obtain this result by showing that the equilibrium measures of an affine iterated function system are never Bernoulli measures unless the system either is reducible or consists of similarity transformations. The proof builds on earlier results in the thermodynamic formalism of affine iterated function systems due to Bochi, Feng, Käenmäki, Shmerkin and the first named author and also relies on the work of Benoist on the spectral properties of Zariski-dense subsemigroups of reductive linear groups.

Certification of incompatible measurements using quantum steering

In this letter we consider the problem of certification of quantum measurements with an arbitrary number of outcomes. We propose a simple scheme for certifying any set of $d$-outcome projective measurements which do not share any common invariant proper subspace, termed here genuinely incompatible, and the maximally entangled state of two qudits. For our purpose, we focus on a simpler scenario, termed as one-sided device-independent scenario where the resource employed for certification is quantum steering. We also study robustness of our self-testing statements for a certain class of genuinely incompatible measurements including mutually unbiased bases which are essential for several quantum information-theoretic tasks such as quantum cryptography.

Multiplication by finite Blaschke factors on a general class of Hardy spaces

A broader class of Hardy spaces and Lebesgue spaces have been introduced recently on the unit circle by considering continuous \(\Vert \cdot \Vert _1\)-dominating normalized gauge norms instead of the classical norms on measurable functions. A Beurling type result has been proved for the operator of multiplication by the coordinate function. In this paper, we generalize the above Beurling type result to the context of multiplication by a finite Blaschke factor B(z) and also derive the common invariant subspaces of \(B^2(z)\) and \(B^3(z).\) These results lead to a factorization result for all functions in the Hardy space equipped with a continuous rotationally symmetric norm.

A canonical representation of maximal, indecomposable semi groups of constant-rank nonnegative r-potent matrices in ݑ0ݑ›(R)

An ݑŸ-potent matrix in ݑ0ݑ›(R) is an ݑ› × ݑ› matrix satisfying ݐذݑŸ = ݐ¸. A multiplicative semigroup S in ݑ0ݑ›(R) is said to be decomposable if there exists a special kind of common invariant subspace called standard invariant subspace for each ݐ´ ∈ S. A semi-group S of non-negative ݑŸ-potent matrices in ݑ0ݑ›(R) is known to be decomposable if rank (ݑ†) > ݑŸ − 1 for all ݑ† in S. Further, a semigroup S in ݑ0ݑ›(R) of nonnegative matrices will be called a full semigroup if S has no common zero row and no common zero column. We have studied the structure of maximal semi-groups of non-negative ݑŸ-potent matrices in ݑ0ݑ›(R) under the special condition of fullness. The objectives of this paper are twofold: (1) To find conditions under which semigroups of nonnegative r-potent matrices can be expressed as a direct sum of maximal rank-one indecomposable semigroups of r-potent matrices; and (2) To obtain a canonical representation of maximal indecomposable rank-one semigroups of r-potent matrices which in the light of the above result gives a complete characterization of such semigroups having constant rank.

A novel procedure for constructing invariant subspaces of a set of matrices

A problem that is frequently encountered in a variety of mathematical contexts is to find the common invariant subspaces of a single or of a set of matrices. A new method is proposed that gives a definitive answer to this problem. The key idea consists of finding common eigenvectors for exterior powers of the matrices concerned. A convenient formulation of the Plücker relations is then used to ensure that these eigenvectors actually correspond to subspaces or provide the initial constraints for eigenvectors involving parameters. A procedure for computing the divisors of a totally decomposable vector is also provided. Several examples are given for which the calculations are too tedious to do by hand and are performed by coding the conditions found into Maple. Our main motivation lies in Lie symmetry, where the invariant subspaces of the adjoint representations for the Lie symmetry algebra of a differential equation must be known explicitly and comprehensively in order to determine all the ideals of the Lie symmetry algebra.

Energy fluctuation relations and repeated quantum measurements

In this paper, we discuss the statistical description in non-equilibrium regimes of energy fluctuations originated by the interaction between a quantum system and a measurement apparatus applying a sequence of repeated quantum measurements. To properly quantify the information about energy fluctuations, both the exchanged heat probability density function and the corresponding characteristic function are derived and interpreted. Then, we discuss the conditions allowing for the validity of the fluctuation theorem in Jarzynski form 〈e−βQ〉=1, thus showing that the fluctuation relation is robust against the presence of randomness in the time intervals between measurements. Moreover, also the late-time, asymptotic properties of the heat characteristic function are analyzed, in the thermodynamic limit of many intermediate quantum measurements. In such a limit, the quantum system tends to the maximally mixed state (thus corresponding to a thermal state with infinite temperature) unless the system’s Hamiltonian and the intermediate measurement observable share a common invariant subspace. Then, in this context, we also discuss how energy fluctuation relations change when the system operates in the quantum Zeno regime. Finally, the theoretical results are illustrated for the special cases of two- and three-levels quantum systems, now ubiquitous for quantum applications and technologies.

Simultaneous block diagonalization of matrices of finite order

It is well known that a set of non-defect matrices can be simultaneously diagonalized if and only if the matrices commute. In the case of non-commuting matrices, the best that can be achieved is simultaneous block diagonalization. Here we give an efficient algorithm to explicitly compute a transfer matrix which realizes the simultaneous block diagonalization of unitary matrices whose decomposition in irreducible blocks (common invariant subspaces) is known from elsewhere. Our main motivation lies in particle physics, where the resulting transfer matrix must be known explicitly in order to unequivocally determine the action of outer automorphisms such as parity, charge conjugation, or time reversal on the particle spectrum.

Subspace Decomposition for Graphon LQR: Applications to VLSNs of Harmonic Oscillators

Graphon control has been proposed and developed in [1]-[3] to approximately solve control problems for very large-scale networks (VLSNs) of linear dynamical systems based on graphon limits. This paper provides a solution method based on invariant subspace decompositions for a class of graphon linear quadratic regulation (LQR) problems where the local dynamics share homogeneous parameters but the graphon couplings may be heterogeneous among the coupled agents. Graphon couplings in this paper appear in states, controls and costs, and these couplings may be represented by different graphons. By exploring a common invariant subspace of the couplings, the original problem is decomposed into a network coupled LQR problem of finite dimension and a decoupled infinite dimensional LQR problem. A centralized optimal solution, and a nodal collaborative optimal control solution where each agent computes its part of the optimal solution locally, are established. The application of these solutions to finite network LQR problems may be via (i) the graphon control methodology [3], or (ii) the representation of finite LQR problems as special cases of graphon LQR problems. The complexity of these solutions involves solving one nd X nd dimensional Riccati equation and one n X n Riccati equation, where n is the dimension of each nodal agent state and d is the dimension of the nontrivial common invariant subspace of the coupling operators, whereas a direct approach involves solving an nN X nN dimensional Riccati equation, where N is the size of the network. For situations where the graphon couplings do not admit exact low-rank representations, approximate control is developed based on low-rank approximations. Finally, an application to the regulation of harmonic oscillators coupled over large networks with uncertainties is demonstrated.

A strongly irreducible affine iterated function system with two invariant measures of maximal dimension

AbstractA classical theorem of Hutchinson asserts that if an iterated function system acts on $\mathbb {R}^{d}$ by similitudes and satisfies the open set condition then it admits a unique self-similar measure with Hausdorff dimension equal to the dimension of the attractor. In the class of measures on the attractor, which arise as the projections of shift-invariant measures on the coding space, this self-similar measure is the unique measure of maximal dimension. In the context of affine iterated function systems it is known that there may be multiple shift-invariant measures of maximal dimension if the linear parts of the affinities share a common invariant subspace, or more generally if they preserve a finite union of proper subspaces of $\mathbb {R}^{d}$ . In this paper we give an example where multiple invariant measures of maximal dimension exist even though the linear parts of the affinities do not preserve a finite union of proper subspaces.

Spectral hypersurfaces for operator pairs and Hadamard matrices of F type

We prove that if for a pair of $$n\times n$$ matrices A and B, with A being normal, the projective joint spectrum of A, B, $$\mathrm{AB}$$ and the identity is given by $$\begin{aligned} \sigma (A,B,\mathrm{AB},I)=\{[x,y,z,t]\in {\mathbb {C}}{\mathbb P}^3:x^n+y^n+(-1)^{n-1}z^n-t^n=0 \}, \end{aligned}$$ then this pair is unitary equivalent to a one associated with a complex Hadamard matrix of order n. If $$n=3,4$$ , or 5, where there is a complete description of Hadamard matrices, we list those that generate a pair with the above mentioned spectrum. If $$\begin{aligned}&\sigma (A,B,\mathrm{AB},\mathrm{BA},I) =\{[x,y,z_1,z_2,t]\in {\mathbb {C}}\mathbb { P}^4: \\&x^n+y^n+(-1)^{n-1}(e^{2\pi i/n}z_1+z_2)^n-t^n=0\}, \end{aligned}$$ this Hadamard matrix is exactly the Fourier matrix $$F_n$$ . If for an operator pair A, B acting on a Hilbert space, such hypersurfaces appear as components of the projective joint spectrum of the corresponding tuples, then under some mild conditions the pair has a common invariant subspace of dimension n, and the restriction of A, B to this subspace is generated by a Hadamard matrix of F type.

Taylor spectra and common invariant subspaces through the Duggal and generalized Aluthge transforms for commuting n-tuples of operators

n the first part of this paper, we introduce two notions of multivariable Duggal transforms (toral and spherical), and study their basic properties including hyponormality and norm-continuity. In the second part, we study how the Taylor spectrum and Taylor essential spectrum of 2-variable weighted shifts behave under the toral and spherical Duggal transforms including generalized Aluthge transforms. In the last part, we investigate nontrivial common invariant subspaces between the toral (respectively spherical) Duggal transform and the original n-tuple of bounded operators with dense ranges. We also study the sets of common invariant subspaces among them.

Common invariant subspaces from small commutators

Common invariant subspaces from small commutators

The common invariant subspace problem and Tarski’s theorem

The common invariant subspace problem and Tarski’s theorem

Tight bound for deciding convergence of consensus systems

We analyze the asymptotic convergence of all infinite products of matrices taken in a given finite set by looking only at finite or periodic products. It is known that when the matrices of the set have a common nonincreasing polyhedral norm, all infinite products converge to zero if and only if all infinite periodic products with period smaller than a certain value converge to zero. Moreover, bounds on that value are available (Lagarias and Wang, 1995).We provide a stronger bound that holds for both polyhedral norms and polyhedral seminorms. In the latter case, the matrix products do not necessarily converge to zero, but all trajectories of the associated system converge to a common invariant subspace. We prove that our bound is tight for all seminorms.Our work is motivated by problems in consensus systems, where the matrices are stochastic (nonnegative with rows summing to one), and hence always share a same common nonincreasing polyhedral seminorm. In that case, we also improve existing results.

The common invariant subspace problem and Tarski’s theorem

This article presents a computable criterion for the existence of a common invariant subspace of $n\times n$ complex matrices $A_{1}, \dots ,A_{s}$ of a fixed dimension $1\leq d\leq n$. The approach taken in the paper is model-theoretic. Namely, the criterion is based on a constructive proof of the renowned Tarski's theorem on quantifier elimination in the theory $\ACF$ of algebraically closed fields. This means that for an arbitrary formula $\varphi$ of the language of fields, a quantifier-free formula $\varphi'$ such that $\varphi\lra\varphi'$ in $\ACF$ is given explicitly. The construction of $\varphi'$ is elementary and based on the effective Nullstellensatz. The existence of a common invariant subspace of $A_{1},\dots,A_{s}$ of dimension $d$ can be expressed in the first-order language of fields, and hence, the constructive version of Tarski's theorem yields the criterion. In addition, some applications of this criterion in quantum information theory are discussed.

Aluthge Transforms and Common Invariant Subspaces for a Commuting $$\varvec{n}$$ n -Tuple of Operators

It is well known that a bounded operator with dense range has a nontrivial invariant subspace if and only if its Aluthge transform does. Recently, R. Curto and Jasang Yoon have introduced the toral and spherical Aluthge transforms for commuting pairs (with particular emphasis on spherically quasinormal and spherically isometric 2-variable weighted shifts) and studied their basic properties. In this paper, we first investigate nontrivial common invariant subspaces between the toral (resp. spherical) Aluthge transform and the original n-tuple of bounded operators with dense ranges. We next study the sets of common invariant subspaces among them. Finally, we give an example which shows that the sets of common invariant subspaces between two Aluthge transforms (toral and spherical) are not isomorphic, even though they are equivalent with respect to the property of existence of a common nontrivial invariant subspace.

Invariant and Almost-Invariant Subspaces for Pairs of Idempotents

It is known that every bounded operator on an infinite dimensional separable Hilbert space \({\mathcal{H}}\) has an invariant subspace if and only if each pair of idempotents on \({\mathcal{H}}\) has a common invariant subspace. We show that the same equivalence holds for operators and pairs of idempotents that are essentially selfadjoint. We also show that each pair of idempotents on \({\mathcal{H}}\) has a common almost-invariant half-space.

Compact noncontraction semigroups of affine operators

We analyze compact multiplicative semigroups of affine operators acting in a finite-dimensional space. The main result states that every such semigroup is either contracting, that is, contains elements of arbitrarily small operator norm, or all its operators share a common invariant affine subspace on which this semigroup is contracting. The proof uses functional difference equations with contraction of the argument. We look at applications to self-affine partitions of convex sets, the investigation of finite affine semigroups and the proof of a criterion of primitivity for nonnegative matrix families. Bibliography: 32 titles.