On triangular similarity of nilpotent triangular matrices

Criterion of unitary similarity for upper triangular matrices in general position

Each square complex matrix is unitarily similar to an upper triangular matrix with diagonal entries in any prescribed order. Let A and B be upper triangular n-by-n matrices that (i) are not similar to direct sums of matrices of smaller sizes, or (ii) are in general position and have the same main diagonal. We prove that A and B are unitarily similar if and only if ||h(A_k)||=||h(B_k)|| for all complex polynomials h(x) and k=1, 2, . . , n, where A_k and B_k are the principal k-by-k submatrices of A and B, and ||M|| is the Frobenius norm of M.

Description of B-orbit closures of order 2 in upper-triangular matrices

Let n_n(C) be the algebra of strictly upper-triangular n x n matrices over the field of complex numbers and X_2 the subset of matrices of nilpotent order 2. Let B_n(C) be the group of invertible upper-triangular matrices acting on n_n(C) by conjugation. Let B(u) be the orbit of u in X_2 with respect to this action. Let S_n^2 be the subset of involutions in the symmetric group S_n. We define a new partial order on S_n^2 which gives the combinatorial description of the closure of B(u). We also construct an ideal I(B(u)) in symmetric algebra S(n_n(C)^* whose variety V(I(B(u))) equals the closure …

Normal forms for orthogonal similarity classes of skew-symmetric matrices

Let F be an algebraically closed field of characteristic different from 2. We show that every nonsingular skew-symmetric n by n matrix X over F is orthogonally similar to a bidiagonal skew-symmetric matrix. In the singular case one has to allow some 4-diagonal blocks as well. If further the characteristic is 0, we construct the normal form for O_n(F)-similarity classes of skew-symmetric matrices. In this case the known normal forms (as presented in the well known book by Gantmacher) are quite different. Finally we study some related varieties of matrices. We prove that the variety of normalize…

图像幂零 η-扩张上的高阶余挠对

Higher Cotorsion Pairs on Image-Nilpotent \(η\)-Extensions

本文针对图像幂零 η-扩张引入一致图像幂零性，证明右 n-余挠对的提升定理与完备性定理，并统一多种环与范畴构造。

非退化矩阵的酉相似性

Unitary Similarity of Nonderogatory Matrices

针对非退化矩阵的酉相似性验证问题，提出一种基于几何方法的稳定且高效的数值解法。

Jordan types of triangular matrices over a finite field

Let $λ$ be a partition of an integer $n$ and ${\mathbb F}_q$ be a finite field of order $q$. Let $P_λ(q)$ be the number of strictly upper triangular $n\times n$ matrices of the Jordan type $λ$. It is known that the polynomial $P_λ$ has a tendency to be divisible by high powers of $q$ and $Q=q-1$, and we put $P_λ(q)=q^{d(λ)}Q^{e(λ)}R_λ(q)$, where $R_λ(0)\neq0$ and $R_λ(1)\neq0$. In this article, we study the polynomials $P_λ(q)$ and $R_λ(q)$. Our main results: an explicit formula for $d(λ)$ (an explicit formula for $e(λ)$ is known, see Proposition 3.3 below), a recursive formula for $R_λ(q)$ (a…

On types and classes of commuting matrices over finite fields

This paper addresses various questions about pairs of similarity classes of matrices which contain commuting elements. In the case of matrices over finite fields, we show that the problem of determining such pairs reduces to a question about nilpotent classes; this reduction makes use of class types in the sense of Steinberg and Green. We investigate the set of scalars that arise as determinants of elements of the centralizer algebra of a matrix, providing a complete description of this set in terms of the class type of the matrix. Several results are established concerning the commuting of ni…

Block triangular miniversal deformations of matrices and matrix pencils

For each square complex matrix, V. I. Arnold constructed a normal form with the minimal number of parameters to which a family of all matrices B that are close enough to this matrix can be reduced by similarity transformations that smoothly depend on the entries of B. Analogous normal forms were also constructed for families of complex matrix pencils by A. Edelman, E. Elmroth, and B. Kagstrom, and contragredient matrix pencils (i.e., of matrix pairs up to transformations (A,B)-->(S^{-1}AR,R^{-1}BS)) by M. I. Garcia-Planas and V. V. Sergeichuk. In this paper we give other normal forms for famil…

Reduction for characters of finite algebra groups

Let J be a finite-dimensional nilpotent algebra over a finite field F_q. We formulate a procedure for analysing characters of the group 1+J. In particular, we study characters of the group $U_n (q)$ of unipotent triangular $n\times n$ matrices over F_q. Using our procedure, we compute the number of irreducible characters of $U_n (q)$ of each degree for n<14. Also, we explain and generalise a phenomenon concerning the group $U_{13}(2)$ discovered by Isaacs and Karagueuzian.

Littlewood's algorithm and quaternion matrices

A strengthened form of Schur's triangularization theorem is given for quaternion matrices with real spectrum (for complex matrices it was given by Littlewood). Littlewood's algorithm for reducing a complex matrix to a canonical form under unitary similarity is extended to quaternion matrices whose eigenvalues have geometric multiplicity 1.

计算类型A_n的ad-nilpotent理想中的轨道品种的F_q点数

Counting $\mathbb F_q$-points of orbital varieties in ad-nilpotent ideals of type $A_n$

本文研究了ad-nilpotent理想中轨道品种的F_q点数，通过Hall标量积和修改的Hall-Littlewood函数与染色准对称函数的乘积，以及标准Young表的显式集合，给出了两种计数公式，并应用于Hessenberg变种、零平方元素和双余类的计数。

矩阵对的多项式相似性

Polynomial similarity of pairs of matrices

本文研究矩阵对的多项式相似性，通过多项式等价与相似性定义该关系，并证明其分类问题是 wild 的，包含经典不可解的矩阵对相似分类问题。