Jordan canonical form of Pascal-type matrices via sequences of binomial type

In this paper, we study the Jordan canonical form of the generalized Pascal functional matrix associated with a sequence of binomial type, and demonstrate that the transition matrix between the generalized Pascal functional matrix and its Jordan canonical form is the iteration matrix associated with the binomial sequence. In addition, some combinatorial identities are derived from the corresponding matrix factorization.     

投射可迁环上矩阵环的若当同态

设R′是一个环,Mn′（R′）是R′上的n′×n′矩阵环.如果环R有不变基数性质并且每个有限生成的投射左R-模是自由模,则R是一个投射自由环.如果环R≌Mr（S）,其中S是一个投射自由环,则R是一个投射可迁环.当R是一个投射可迁环时,给出了从Mn′（R′）到Mn（R）（n′≥n≥2）的若当同态的代数公式.     

On a divisor of the central binomial coefficient

The Ramanujan Journal - It is well known that for all $$n\ge 1$$ the number $$n+1$$ is a divisor of the central binomial coefficient $${2n\atopwithdelims ()n}$$ . Since the nth central binomial...     

求矩阵A的Jordan标准形的另一方法

本提出了求矩阵A的Jordan标准形的另一方法：利用rank(λ(E-A)^P的结果，得出了对应于特征(λi的Jordan块的阶数和个数，然后求出矩阵A的Jordan标准形．     

Condition number of the matrix of transition to the normal Jordan form

We establish necessary and sufficient conditions for the well-conditioned reduction of a matrix to the Jordan normal form. ZBViK Plant, Kamenets-Podol'skii. Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 51, No. 1, pp. 120–122, January, 1999.     

Determinant of a matrix that commutes with a Jordan matrix

Given a Jordan matrix J, we obtain an explicit formula for the determinant of any matrix T that commutes with it.     

Solution of linear equations with coefficient matrix in band form

An algorithm is given for the solution of simultaneous linear equations having a band type coefficient matrix. The algorithm utilizes a suitable partitioning of the coefficient matrix. Some particular cases are described in which the use of the algorithm is advisable.This research was supported in part by the National Aeronautics and Space Administration Grant No. NGR-33-015-013.     

Generalized Jordan matrix of a linear operator

For any linear operator defined over an arbitrary field k, there is a basis in which this matrix is a generalized Jordan matrix (of the second kind) with elements in the field k. For any linear operator, such a matrix is defined uniquely up to permutation of diagonal blocks.     

On the Jordan form in a unitary space

A method is given, which determines an orthonormal part of a basis with reto which a given transformation is in Jordan canonical form.     

Number of divisors of the central binomial coefficient

Asymptotic formulas are derived for the following expressions: log τ(C _2n ⁿ ) and log τ([1, ... , n]).     

Irreducible modules over a Jordan plane over a field of characteristic zero

Irreducible modules over a Jordan plane over a field of characteristic zero are considered. All finite-dimensional modules are described. Examples of infinite-dimensional irreducible modules are constructed.