Formal description,compression and transformation of digital pictures

In an earlier paper (Part I) we described the construction of Hermite code for multiple grey-level pictures using the concepts of vector spaces over Galois Fields. In this paper a new algebra is worked out for Hermite codes to devise algorithms for various transformations such as translation, reflection, rotation, expansion and replication of the original picture. Also other operations such as concatenation, complementation, superposition, Jordan-sum and selective segmentation are considered. It is shown that the Hermite code of a picture is very powerful and serves as a mathematical signature of the picture. The Hermite code will have extensive applications in picture processing, pattern recognition and artificial intelligence.     

有强法式的体上矩阵

设D为除环,A∈Dn×n,则可用初等变换将λI-A化简为对角阵A= diag(1,…,1,φ1,…,φr),其中(?)i为D上首1多项式并且φ1|…|φr.如果这个对角阵A在形状上是唯一的,则称A是有强法式的矩阵.本文应用中心原子因子与初等因子给出了体上有强法式的矩阵的本质刻画,给出了体上矩阵有强法式的一些充要条件.     

紧致交换矩阵半群

通过将矩阵同时对角化或同时上三角化的方法,给出有关紧致Abel矩阵半群以及紧致Hermite矩阵半群中矩阵的特征值的一些很好的刻画,证明了由可逆的Hermite矩阵构成的紧致矩阵半群中每个矩阵的特征值都是±1,Hermite矩阵单半群相似于对角矩阵半群,紧致交换矩阵半群的谱半径不超过1,等等.     

Solving confluent Vandermonde systems of Hermite type

In this paper the authors develop two algorithms to solve systems of linear equations where the coefficients form a confluent Vandermonde matrix of Hermite type, or its transpose. These algorithms reduce the given system to upper triangular form by means of elementary matrix transformations. Recursive formulas to obtain the upper triangular form in an economical way are derived. Applications and numerical results are included.Algol-60 programs are appended.     

Similarity transformations of decomposable matrix polynomials and related questions

A relationship is found between the similarity transformations of decomposable matrix polynomials with relatively prime elementary divisors and the equivalence transformations of the corresponding matrices with scalar entries. Matrices with scalar entries are classified with respect to equivalence transformations based on direct sums of lower triangular almost Toeplitz matrices. This solves the similarity problem for a special class of finite matrix sets over the field of complex numbers. Eventually, this problem reduces to the one of special diagonal equivalence between matrices. Invariants of this equivalence are found.     

Generalization of Mirsky’s theorem on diagonals and eigenvalues of matrices

Mirsky proved that, for the existence of a complex matrix with given eigenvalues and diagonal entries, the obvious necessary condition is also sufficient. We generalize this theorem to matrices over any field and provide a short proof. Moreover, we show that there is a unique companion-matrix-type solution for this problem.     

特征值反问题的唯一性

证明了由特征值及特征向量反求矩阵时,特征值在对角矩阵中的排序可以是任意的,只须将对应特征向量作相应排序,所得矩阵唯一。对于重特征值的线性无关的特征向量可任意选取,所得矩阵唯一。     

The invariance of elementary symmetric functions

The general problem considered is: what linear transformations on matrices preserve certain prescribed invariants or other properties of the matrices? Specifically, the forms of the following linear transformations are determined: the linear transformations that hold either the trace or the second elementary symmetric function of the eigenvalues of each matrix fixed, and in addition preserve either the determinant, or the permanent, or an elementary symmetric function of the squares of the singular values, or the property of being a rank 1 matrix or a unitary matrix.     

Generating equations approach for quadratic matrix equations

We show how Van Loan's method for annulling the (2,1) block of skew‐Hamiltonian matrices by symplectic‐orthogonal similarity transformation generalizes to general matrices and provides a numerical algorithm for solving the general quadratic matrix equation: For skew‐Hamiltonian matrices we find their canonical form under a similarity transformation and find the class of all symplectic‐orthogonal similarity transformations for annulling the (2,1) block and simultaneously bringing the (1,1) block to Hessenberg form. We present a structure‐preserving algorithm for the solution of continuous‐time algebraic Riccati equation. Unlike other methods in the literature, the final transformed Hamiltonian matrix is not in Hamiltonian–Schur form. Three applications are presented: (a) for a special system of partial differential equations of second order for a single unknown function, we obtain the matrix of partial derivatives of second order of the unknown function by only algebraic operations and differentiation of functions; (b) for a similar transformation of a complex matrix into a symmetric (and three‐diagonal) one by applying only finite algebraic transformations; and (c) for finite‐step reduction of the eigenvalues–eigenvectors problem of a Hermitian matrix to the eigenvalues– eigenvectors problem of a real symmetric matrix of the same dimension. Copyright © 1999 John Wiley & Sons, Ltd.     

A quaternion QR algorithm

Summary This paper extends the Francis QR algorithm to quaternion and antiquaternion matrices. It calculates a quaternion version of the Schur decomposition using quaternion unitary similarity transformations. Following a finite step reduction to a Hessenberg-like condensed form, a sequence of implicit QR steps reduces the matrix to triangular form. Eigenvalues may be read off the diagonal. Eigenvectors may be obtained from simple back substitutions. For serial computation, the algorithm uses only half the work and storage of the unstructured Francis QR iteration. By preserving quaternion structure, the algorithm calculates the eigenvalues of a nearby quaternion matrix despite rounding errors.     

Hermite正定对称矩阵迹的一些结果(英文)

本文研究了一类Hermite正定矩阵迹的不等式问题.利用文献[2-6]中的结果以及放缩法,获得了Hermite正定矩阵迹的极值定理、杨氏不等式和贝努利不等式,并且将许多初等不等式推广到Hermite正定矩阵迹的情形.     

Trace preserving linear transformations on matrix algebras

A characterization is given for all linear transformations on a matrix algebra which preserve (1) the trace (2) the unit matrix (3) the trace and the second elementary symmetric function of the eigenvalues.     

体上代数矩阵的素有理标准形与特征值

本文应用最小中心多项式给出了体上代数矩阵的素有理标准形与初等因子组的构造，由此得到体上代数矩阵相似的充要条件以及相似于一个对角矩阵的充要条件．本文还讨论了体上矩阵的左、右特征值的构造与性质．     

求解矩阵方程AXB+CXD=F参数迭代法的最优参数分析

本文针对求矩阵方程AXB+CXD=F唯一解的参数迭代法，分析当矩阵A，B，C，D均是Hermite正（负）定矩阵时，迭代矩阵的特征值表达式，给出了最优参数的确定方法，并提出了相应的加速算法.     

Semiscalar equivalence and the factorization of polynomial matrices

One considers the problem of the factorization of polynomial matrices over an arbitrary field in connection with their reducibility by semiscalar equivalent transformations to triangular form with the invariant factors along the principal diagonal. In particular, one establishes a criterion for the representability of a polynomial matrix in the form of a product of factors (the first of which is unital), the product of the canonical diagonal forms of which is equal to the canonical diagonal form of the given matrix. There is given also a method for the construction of such factorizations.Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 42, No. 5, pp. 644–649, May, 1990.     

Hadamard定理在四元数体上的推广

Let K be any skew-field with central field F. A matrix A=(a_ij)n×n over K is called centralized if the characteristic matrix λI-A can be reduced by some elementary transformations into the following diagonal form:such that are all manic polynomials over F. The determinant of a centralized matrix A=(a_ij)n×n may be defined by (1) asfollows: and then the famous theorem of Hadamard can be generalized as in the following: Theorem. If A=(a_ij)n×n is a non-singular centralized matrix over the skewfield of quaternions, thenand the equality sign holds if and only if the columns of A are all muturally orthogonal. This theorem may be proved by the following three lemmas: Lemma 1. If A is a centralized n-rowed square matrix of quaternions, then sois and Lemma 2. For any n-rowed square, matrix A of quaternions, the. following two matrices are always centralized:and Lemma 3. If A is a centralized matrix of quaternions, then we have     

Numerical Determination of a Canonical Form of a Symplectic Matrix

We propose an algorithm that transforms a real symplectic matrix with a stable structure to a block diagonal form composed of three main blocks. The two extreme blocks of the same size are associated respectively with the eigenvalues outside and inside the unit circle. Moreover, these eigenvalues are symmetric with respect to the unit circle. The central block is in turn composed of several diagonal blocks whose eigenvalues are on the unit circle and satisfy a modification of the Krein-Gelfand-Lidskii criterion. The proposed algorithm also gives a qualitative criterion for structural stability.     

On the hermite matrix of a polynomial

We investigate the location of the eigenvalues of the Hermite matrix of a given complex polynomial, the question under what conditions a given polynomial and the characteristic polynomial of its Hermite matrix are identical, and the question under what conditions the Hermite matrix has only one distinct eigenvalue.     

On a class of Jacobi-like procedures for diagonalising arbitrary real matrices

Summary A new class of elementary matrices is presented which are convenient in Jacobi-like diagonalisation methods for arbitrary real matrices. It is shown that the presented transformations possess the normreducing property and that they produce an ultimate quadratic convergence even in the case of complex eigenvalues. Finally, a quadratically convergent Jacobi-like algorithm for real matrices with complex eigenvalues is presented.     

The retraction algorithm for factoring banded symmetric matrices

Let A be an n × n symmetric matrix of bandwidth 2m + 1. The matrix need not be positive definite. In this paper we will present an algorithm for factoring A which preserves symmetry and the band structure and limits element growth in the factorization. With this factorization one may solve a linear system with A as the coefficient matrix and determine the inertia of A, the number of positive, negative, and zero eigenvalues of A. The algorithm requires between 1/2nm² and 5/4nm² multiplications and at most (2m + 1)n locations compared to non‐symmetric Gaussian elimination which requires between nm² and 2nm² multiplications and at most (3m + 1)n locations. Our algorithm reduces A to block diagonal form with 1 × 1 and 2 × 2 blocks on the diagonal. When pivoting for stability and subsequent transformations produce non‐zero elements outside the original band, column/row transformations are used to retract the bandwidth. To decrease the operation count and the necessary storage, we use the fact that the correction outside the band is rank‐1 and invert the process, applying the transformations that would restore the bandwidth first, followed by a modified correction. This paper contains an element growth analysis and a computational comparison with LAPACKs non‐symmetric band routines and the Snap‐back code of Irony and Toledo. Copyright © 2007 John Wiley & Sons, Ltd.