Block Triangular Matrices Research Articles

Counting gradings on Lie algebras of block-triangular matrices

We study the number of isomorphism classes of gradings on Lie algebras of block-triangular matrices. Let G be a finite abelian group, for m∈Ns we determine the number E(−)(G,m) of isomorphism classes of elementary G-gradings on the Lie algebra UT(m)(−) of block-triangular matrices over an algebraically closed field of characteristic zero. We study the asymptotic growth of E(−)(G,m) and as a consequence prove that the E(−)(G,⋅) determines G up to isomorphism. We also study the asymptotic growth of the number N(−)(G,m) of isomorphism classes of G-gradings on UT(m)(−) and prove that N(−)(G,m))∼|G|E(−)(G,m).

Gradings on block-triangular matrix algebras

Upper triangular, and more generally, block-triangular matrices, are rather important in Linear Algebra, and also in Ring theory, namely in the theory of PI algebras (algebras that satisfy polynomial identities). The group gradings on such algebras have been extensively studied during the last decades. In this paper we prove that for any group grading on a block-triangular matrix algebra, over an arbitrary field, the Jacobson radical is a graded (homogeneous) ideal. As noted by F. Yasumura [Arch. Math. (Basel) 110 (2018), pp. 327–332] this yields the classification of the group gradings on these algebras and confirms a conjecture made by A. Valenti and M. Zaicev [Arch. Math. (Basel) 89 (2007), pp. 33–40].

Upper triangular matrices of graded division algebras and their identities

In the present paper we study UT(D1,…,Dn), a G-graded algebra of block triangular matrices where G is a group and the diagonal blocks D1,…,Dn are graded division algebras. We prove that any two such algebras are G-isomorphic if and only if they satisfy the same graded polynomial identities. We also discuss the number of different isomorphism classes obtained by varying the grading and we exhibit its connection with the factorability of the T-ideal of graded identities. Moreover we give some results about the generators of the graded polynomial identities for these algebras. In particular we generalize the results about the graded identities of UTn to the case in which the diagonal blocks D1,…,Dn are all isomorphic.

Robot stiffness theory reconsideration based on Schur complement eigenvalues: Extension to GSP dynamic stiffness evaluation

The known robot stiffness evaluation algorithm based on the Schur complements’ eigenvalues (SCE) is refined and its application is extended to the dynamic stiffness problem. For the refinement, the Schur complements synthesis is reduced to standard transformations of the stiffness matrix into the triangular block matrices. In these terms, the Schur complements of the robot dynamic stiffness matrix are formulated for steady-state forced sinusoidal vibrations of a damping-free system. The interrelationships between the forced-frequency-dependent SCE and natural frequencies are established: the frequencies resulting in the SCE zero values are proven to be identical to the natural frequencies, and vice versa. The SCE of the dynamic stiffness matrix defines the range and extremes of the dynamic stiffness values that allow the building of the translational and rotational stiffness ellipsoids and formulation of the stiffness-related robot performance indices. In four application examples, the SCE-based dynamic stiffness values of the Gough-Stewart platform (GSP) on weightless supports are defined. The definitions allow the formulation of the dynamic stiffness-related performance indices which are used for quantitative evaluation and comparison of the GSP designs.

Graded identities of Mn(E) and their generalizations over infinite fields

Let G be a group and F an infinite field. Assume that A is a finite dimensional F-algebra with an elementary G-grading. In this paper, we study the graded identities satisfied by the tensor product grading on the F-algebra A⊗C, where C is an H-graded colour β-commutative algebra. More precisely, under a technical condition, we provide a basis for the TG-ideal of graded polynomial identities of A⊗C, up to graded monomial identities. Furthermore, the F-algebra of upper block-triangular matrices UT(d1,…,dn), as well as the matrix algebra Mn(F), with an elementary grading such that the neutral component corresponds to its diagonal, are studied. As a consequence of our results, a basis for the graded identities, up to graded monomial identities of degrees ≤2d−1, for Md(E) and Mq(F)⊗UT(d1,…,dn), with a tensor product grading, is exhibited. In this latter case, d=d1+…+dn. Here E denotes the infinite dimensional Grassmann algebra with its natural Z2-grading, and the grading on Mq(F) is Pauli grading. The results presented in this paper generalize results from [14] and from other papers which were obtained for fields of characteristic zero.

Existence of the solution of the Sylvester-type matrix equation in the ring of block triangular matrices

Existence of the solution of the Sylvester-type matrix equation in the ring of block triangular matrices

Centralizing additive maps on rank r block triangular matrices

Let F be a field and let k, n1, . . . , nk be positive integers with n1 + · · · + nk = n > 2. We denote by Tn1,...,nk a block triangular matrix algebra over F with unity In and center Z(Tn1,...,nk ). Fixing an integer 1 1 upper triangular matrices over an arbitrary field is addressed.

On the number of gradings on matrix algebras

We determine the number of isomorphism classes of elementary gradings by a finite group on an algebra of upper block-triangular matrices. As a consequence we prove that, for a finite abelian group G, the sequence of the numbers E(G,m) of isomorphism classes of elementary G-gradings on the algebra Mm(F) of m×m matrices with entries in a field F characterizes G. A formula for the number of isomorphism classes of gradings by a finite abelian group on an algebra of upper block-triangular matrices over an algebraically closed field, with mild restrictions on its characteristic, is also provided. Finally, if G is a finite abelian group, F is an algebraically closed field and N(G,m) is the number of isomorphism classes of G-gradings on Mm(F) we prove that N(G,m)∼1|G|!m|G|−1∼E(G,m).

Triangulations of Operators with Two-Isometric Liftings

The paper deals with bounded linear operators T on a complex Hilbert space $${\mathcal {H}}$$ which have 2-isometric liftings S on a Hilbert space $${\mathcal {K}}$$ containing $${\mathcal {H}}$$ as a closed subspace. We investigate three types of such liftings, and for each type we describe in detail the corresponding operators T by the $$2\times 2$$ upper triangular block matrices. The entries of those matrices are expressed by operators which belong to certain classes of A-contractions for some positive operators A on $${\mathcal {H}}$$ . We also refer to the expansive operators with such liftings, as well as to Brownian isometric or quasi-Brownian unitary liftings.

Optimized Cramer’s Rule in WZ Factorization and Applications

In this paper, W Z factorization is optimized with a proposed Cramer’s rule and compared with classical Cramer’s rule to solve the linear systems of the factorization technique. The matrix norms and performance time of WZ factorization together with LU factorization are analyzed using sparse matrices on MATLAB via AMD and Intel processor to deduce that the optimized Cramer’s rule in the factorization algorithm yields accurate results than LU factorization and conventional W Z factorization. In all, the matrix group and Schur complement for every Zsystem (2×2 block triangular matrices from Z-matrix) are established.

Approximation of pseudospectra of block triangular matrices

We approximate the pseudospectra of a block triangular matrix with two blocks on the diagonal in terms of the pseudospectra of its diagonal blocks.

On the minus partial order in control systems

In this paper, the minus matrix partial order is considered to introduce the concept of minus partial ordered control systems. The transmission of the reachability property under this binary relation is investigated. Furthermore, the analysis of compartmental systems leads us to consider block triangular matrices. Hence, the existence and computation of partially ordered matrices having a similar block structure are studied. These results are applied to compartmental systems to get, via feedback, related systems with the same block structure and ordered under the minus partial order.

Substitution algorithms for rational matrix equations

We study equations of the form $r(X) = A$, where $r$ is a rational function and $A$ and $X$ are square matrices of the same size. We develop two techniques for solving these equations by inverting, through a substitution strategy, two schemes for the evaluation of rational functions of matrices. For block triangular matrices, the new methods yield the same computational cost as the evaluation schemes from which they are obtained. A general equation is reduced to block upper triangular form by exploiting the Schur decomposition of the given matrix. For real data, using the real Schur decomposition, the algorithms compute the real solutions using only real arithmetic, and numerical experiments show that our implementations are superior, in terms of both accuracy and speed, to existing alternatives. Finally, we discuss how these techniques can be used as building blocks in algorithms for computing functions of matrices defined through matrix equation of the type $f(X) = A$, where $f$ is a primary matrix function.

Group gradings on the Lie and Jordan algebras of block-triangular matrices

We classify up to isomorphism all gradings by an arbitrary group G on the Lie algebras of zero-trace upper block-triangular matrices over an algebraically closed field of characteristic 0. It turns out that the support of such a grading always generates an abelian subgroup of G.Assuming that G is abelian, our technique also works to obtain the classification of G-gradings on the upper block-triangular matrices as an associative algebra, over any algebraically closed field. These gradings were originally described by A. Valenti and M. Zaicev in 2012 (assuming characteristic 0 and G finite abelian) and classified up to isomorphism by A. Borges et al. in 2018.Finally, still assuming that G is abelian, we classify G-gradings on the upper block-triangular matrices as a Jordan algebra, over an algebraically closed field of characteristic 0. It turns out that, under these assumptions, the Jordan case is equivalent to the Lie case.

Graded identities and isomorphisms on algebras of upper block-triangular matrices

Let G be an abelian group and K an algebraically closed field of characteristic zero. A. Valenti and M. Zaicev described the G-gradings on upper block-triangular matrix algebras provided that G is finite. We prove that their result holds for any abelian group G: any grading is isomorphic to the tensor product A⊗B of an elementary grading A on an upper block-triangular matrix algebra and a division grading B on a matrix algebra. We then consider the question of whether graded identities A⊗B, where B is an algebra with a division grading, determine A⊗B up to graded isomorphism. In our main result, Theorem 3, we reduce this question to the case of elementary gradings on upper block-triangular matrix algebras which was previously studied by O.M. Di Vincenzo and E. Spinelli.

On the controllability properties of bilinear systems with discrete time associated with block triangular and permutation matrices

On the controllability properties of bilinear systems with discrete time associated with block triangular and permutation matrices

On the Properties of Controllability of Discrete-Time Bilinear Systems Associated with Block-Triangular and Permutation Matrices

The new results associated with properties of the controllability of discrete-time bilinear systems are presented. The properties of the controllability of a class of bilinear systems with block-triangular matrices are discussed. As a consequence, systems with permutation matrices and scalar bounded control are investigated. A series of necessary controllability conditions related to the dimension of the state vector and spectral properties of systems’ matrices are established and proved.

On common α-scalar Lyapunov solutions

We present two new characterizations regarding common α-scalar solutions for systems of Lyapunov matrix inequalities so as to bridge and broaden relevant existing results. As applications of such results, we further explore the cases involving a set of conformally partitioned block triangular matrices and a set of Z-matrices, respectively, to obtain new results concerning these cases, which are useful in the areas of control and systems theory and engineering.

Group gradings on upper block triangular matrices

It was proved by Valenti and Zaicev, in 2011, that, if $G$ is an abelian group and $K$ is an algebraically closed field of characteristic zero, then any $G$-grading on the algebra of upper block triangular matrices over $K$ is isomorphic to a tensor product $M_n(K)\otimes UT(n_1,n_2,\ldots,n_d)$, where $UT(n_1,n_2,\ldots,n_d)$ is endowed with an elementary grading and $M_n(K)$ is provided with a division grading. In this paper, we prove the validity of the same result for a non necessarily commutative group and over an adequate field (characteristic either zero or large enough), not necessarily algebraically closed.

Incremental computation of block triangular matrix exponentials with application to option pricing

We study the problem of computing the matrix exponential of a block triangular matrix in a peculiar way: Block column by block column, from left to right. The need for such an evaluation scheme arises naturally in the context of option pricing in polynomial diffusion models. In this setting a discretization process produces a sequence of nested block triangular matrices, and their exponentials are to be computed at each stage, until a dynamically evaluated criterion allows to stop. Our algorithm is based on scaling and squaring. By carefully reusing certain intermediate quantities from one step to the next, we can efficiently compute such a sequence of matrix exponentials.