Equality conditions for the singular values of 3 × 3 matrices with one-point spectrum

Let Γ_a be an upper triangular 3 × 3 matrix with diagonal entries equal to a complex scalar a. Necessary and su.cient conditions are found for two of the singular values of Γ_a to be equal. These conditions are much simpler than the equality discr ? = 0, where the expression in the left-hand side is the discriminant of the characteristic polynomial ? of the matrix _Ga = Γ_aΓ_a. Understanding the behavior of singular values of Γ_a is important in the problem of finding a matrix with a triple zero eigenvalue that is closest to a given normal matrix A.     

On a nonnegative irreducible matrix that is similar to a positive matrix

Let A be an n×n irreducible nonnegative (elementwise) matrix. Borobia and Moro raised the following question: Suppose that every diagonal of A contains a positive entry. Is A similar to a positive matrix? We give an affirmative answer in the case n = 4.     

Decomposition of Jordan Automorphisms of Strictly Upper Triangular Matrix Algebra Over Commutative Rings

Over a 2-torsionfree commutative ring R with identity, the algebra of all strictly upper triangular n + 1 by n + 1 matrices is denoted by n ₁. In this article, we prove that any Jordan automorphism of n ₁ can be uniquely decomposed as a product of a graph automorphism, a diagonal automorphism, a central automorphism and an inner automorphism for n ≥ 3. In the cases n = 1, 2, we also give a decomposition for any Jordan automorphism of n ₁.     

Comparison of volumes of convex bodies in real, complex, and quaternionic spaces

The classical Busemann-Petty problem (1956) asks, whether origin-symmetric convex bodies in Rn with smaller hyperplane central sections necessarily have smaller volumes. It is known, that the answer is affirmative if n?4 and negative if n>4. The same question can be asked when volumes of hyperplane sections are replaced by other comparison functions having geometric meaning. We give unified analysis of this circle of problems in real, complex, and quaternionic n-dimensional spaces. All cases are treated simultaneously. In particular, we show that the Busemann-Petty problem in the quaternionic n-dimensional space has an affirmative answer if and only if n=2. The method relies on the properties of cosine transforms on the unit sphere. We discuss possible generalizations.     

A graph theoretic approach to matrix inversion by partitioning

LetM be a square matrix whose entries are in some field. Our object is to find a permutation matrixP such thatPM P ^–1 is completely reduced, i.e., is partitioned in block triangular form, so that all submatrices below its diagonal are 0 and all diagonal submatrices are square and irreducible. LetA be the binary (0, 1) matrix obtained fromM by preserving the 0's ofM and replacing the nonzero entries ofM by 1's. ThenA may be regarded as the adjacency matrix of a directed graphD. CallD strongly connected orstrong if any two points ofD are mutually reachable by directed paths. Astrong component ofD is a maximal strong subgraph. Thecondensation D ^* ofD is that digraph whose points are the strong components ofD and whose lines are induced by those ofD. By known methods, we constructD ^* from the digraph,D whose adjacency matrixA was obtained from the original matrixM. LetA ^* be the adjacency matrix ofD ^*. It is easy to show that there exists a permutation matrixQ such thatQA ^* Q ^–1 is an upper triangular matrix. The determination of an appropriate permutation matrixP from this matrixQ is straightforward.This was an informal talk at the International Symposium on Matrix Computation sponsored by SIAM and held in Gatlinburg, Tennessee, April 24–28, 1961 and was an invited address at the SIAM meeting in Stillwater, Oklahoma on August 31, 1961     

On general summability factor theorem of infinite series

A lower triangular matrix with nonzero principal diagonal entries is called a triangle. In this paper we obtain the sufficient conditions for ∑a_nλ_n to be summable ∣A∣_k whenever ∑a_n is summable ∣T∣_k for a triangle T.     

The LU-factorization of totally positive matrices

An n × n real matrix A is an STP (strictly totally positive) matrix if all its minors are strictly positive. An n × n real triangular matrix A is a ΔSTP matrix if all its nontrivial minors are strictly positive. It is proved that A is an STP matrix iff A = LU where L is a lower triangular matrix, U is an upper triangular matrix, and both L and U are ΔSTP matrices. Several related results are proved. These results lead to simple proofs of many of the determinantal properties of STP matrices.     

A compact updating formula for quasi-Newton minimization algorithms

Quasi-Newton algorithms minimize a functionF(x),x ∈R ⁿ, searching at any iterationk along the directions _k=?H _kg_k, whereg _k=?F(x _k) andH _k approximates in some sense the inverse Hessian ofF(x) atx _k. When the matrixH is updated according to the formulas in Broyden's family and when an exact line search is performed at any iteration, a compact algorithm (free from the Broyden's family parameter) can be conceived in terms of the followingn ×n matrix: $$H{_R} = H - Hgg{^T} H/g{^T} Hg,$$ which can be viewed as an approximating reduced inverse Hessian. In this paper, a new algorithm is proposed which uses at any iteration an (n?1)×(n?1) matrixK related toH _R by $$H_R = Q\left[ {\begin{array}{*{20}c} 0 & 0 \\ 0 & K \\ \end{array} } \right]Q$$ whereQ is a suitable orthogonaln×n matrix. The updating formula in terms of the matrixK incorporated in this algorithm is only moderately more complicated than the standard updating formulas for variable-metric methods, but, at the same time, it updates at any iteration a positive definite matrixK, instead of a singular matrixH _R. Other than the compactness with respect to the algorithms with updating formulas in Broyden's class, a further noticeable feature of the reduced Hessian algorithm is that the downhill condition can be stated in a simple way, and thus efficient line searches may be implemented.     

四元数矩阵的广义Schmidt分解与广酉空间中向量组的广义标准正交化

推广了四元数矩阵的Schmidt分解及广酉空间中向量组的标准正交化问题，给出了实四元数矩阵分解为广酉矩阵与生对角元全正的上三角阵乘积的实用方法．     

Finding Well-Conditioned Similarities to Block-Diagonalize Nonsymmetric Matrices Is NP-Hard

Given an upper triangular matrix A ∈ R^n×n and a tolerance τ, we show that the problem of finding a similarity transformation G such that G⁻¹AG is block diagonal with the condition number of G being at most τ is NP-hard. Let ƒ(n) be a polynomial in n. We also show that the problem of finding a similarity transformation G such that G⁻¹AG is block-diagonal with the condition number of G being at most ƒ(n) times larger than the smallest possible is NP-hard.     

On the Cholesky factorization of the Gram matrix of locally supported functions

Cholesky factorization of bi-infinite and semi-infinite matrices is studied and in particular the following is proved. If a bi-infinite matrixA has a Cholesky factorization whose lower triangular factorL and its lower triangular inverse decay exponentially away from the diagonal, then the semi-infinite truncation ofA has a lower triangular Cholesky factor whose elements approach those ofL exponentially. This result is then applied to studying the asymptotic behavior of splines obtained by orthogonalizing a large finite set of B-splines, in particular identifying the limiting profile when the knots are equally spaced.The first and second authors were partially supported by Nato Grant #920209, the second author also by the Alexander von Humboldt Foundation, and the last two authors by the Italian Ministry of University and Scientific and Technological Research.     

Reducing complex matrices to condensed forms by unitary congruence transformations

We show that any n × n conjugate-normal matrix can be brought by a unitary congruence transformation to block-tridiagonal form with the orders of the consecutive diagonal blocks not exceeding 1, 2, 3, ..., respectively. The proof is constructive; namely, a finite process is described that implements the reduction to the desired form. Sufficient conditions are indicated for the orders of the diagonal blocks to stabilize. In this case, the condensed form is a band matrix.     

Derivations of the Lie algebra of infinite strictly upper triangular matrices over a commutative ring

Let 𝒩(∞,R) be the Lie algebra of infinite strictly upper triangular matrices over a commutative ring R. We show that every derivation of 𝒩(∞,R) is a sum of diagonal and inner derivations.     

A fast method to block-diagonalize a Hankel matrix

In this paper, we consider an approximate block diagonalization algorithm of an n×n real Hankel matrix in which the successive transformation matrices are upper triangular Toeplitz matrices, and propose a new fast approach to compute the factorization in O(n ²) operations. This method consists on using the revised Bini method (Lin et al., Theor Comp Sci 315: 511–523, 2004). To motivate our approach, we also propose an approximate factorization variant of the customary fast method based on Schur complementation adapted to the n×n real Hankel matrix. All algorithms have been implemented in Matlab and numerical results are included to illustrate the effectiveness of our approach.     

交换环上的严格上三角矩阵代数上的Lie导子

设R是任意含单位元的交换环,N(R)为R上(n+1)×(n+1)严格上三角矩阵构成的代数．本文证明了当n≥3且2是R的单位时,N(R)上任意Lie导子D可以唯一的表示为D=D_d+D_b+D_c+D_x,其中D_d,D_b,D_c,D_x分别是N(R)上的对角,极端,中心和内Lie导子,在n=2的情况,我们也证明了N(R)上任意Lie导子D可以表示为对角,极端,内Lie导子的和。     

The spectrum is continuous on the set of quasi-n-hyponormal operators

In this paper it is shown that the spectrum σ, a set-valued function, is continuous when the function is restricted to the set of all ‘quasi-n-hyponormal’ operators acting on an infinite-dimensional separable Hilbert space, where a quasi-n-hyponormal operator is defined to be unitarily equivalent to an n×n upper triangular operator matrix whose diagonal entries are hyponormal operators.     

Good Gradings on Upper Block Triangular Matrix Algebras

We investigate group gradings of upper block triangular matrix algebras over a field such that all the matrix units lying there are homogeneous elements. We describe these gradings as endomorphism algebras of graded flags and classify them as orbits of a certain biaction of a Young subgroup and the group G on the set G ⁿ , where G is the grading group and n is the size of the matrix algebra. In particular, the results apply to algebras of upper triangular matrices.     

Real elements in T n (K)

We study the real elements in triangular matrix groups. We describe some classes of elements that are real in T _n (K) – the groups of upper triangular matrices over a commutative field K. From the obtained results there follow some applications for finding real elements in general linear groups – GL _n (K).     

上三角矩阵的线性交换映射的表示

设Trn(R)表示定义在实数域R上的n×n阶上三角矩阵的集合,φ是定义Trn(R)上线性映射.如果对任意X∈Trn(R)有Xφ(X)=φ(X)X成立,称φ是线性交换映射.本文利用初等的矩阵计算方法描述了当φ(I)=I时,线性交换映射φ的表示形式,而且给出了φ的Frobenius范数‖φ(X)‖F的估计.