计算对称正定矩阵逆主$p$-次根的通用算法

On the minima of positive definite binary hamiltonian forms

Let $A$ be a definite quaternion algebra over $\mathbb Q$, with discriminant $D_A$, and $O$ a maximal order of $A$. We show that the minimum of the positive definite hamiltonian binary forms over $O$ with discrimiminant $-1$ is $\sqrt{D_A}$. When the different of $O$ is principal, we provide an explicit form representing this minimum, and when $O$ is principal, we give the list of the equivalence classes of all such forms. We also give criteria and algorithms to determine when the different of $O$ is principal.

Bounded Ratios of Products of Principal Minors of Positive Definite Matrices

Considered is the multiplicative semigroup of ratios of products of principal minors bounded over all positive definite matrices. A long history of literature identifies various elements of this semigroup, all of which lie in a sub-semigroup generated by Hadamard-Fischer inequalities. Via cone-theoretic techniques and the patterns of nullity among positive semidefinite matrices, a semigroup containing all bounded ratios is given. This allows the complete determination of the semigroup of bounded ratios for 4-by-4 positive definite matrices, whose 46 generators include ratios not implied by Had…

Square roots of H-nonnegative matrices

Roots of matrices are well-studied. The conditions for their existence are understood: The block sizes of nilpotent Jordan blocks, arranged in pairs, have to satisfy some simple algebraic property. More interesting are structured roots of structured matrices. Probably the best known example is the existence and uniqueness of positive definite square roots of a positive definite matrix. If one drops the requirement of positive definiteness of the square root, it turns out that there exists an abundance of square roots. Here a description of all canonical forms of all square roots is possible an…

Symmetrizable intersection matrices and their root systems

In this paper we study symmetrizable intersection matrices, namely generalized intersection matrices introduced by P. Slodowy such that they are symmetrizable. Every such matrix can be naturally associated with a root basis and a Weyl root system. Using $d$-fold affinization matrices we give a classification, up to braid-equivalence, for all positive semi-definite symmetrizable intersection matrices. We also give an explicit structure of the Weyl root system for each $d$-fold affinization matrix in terms of the root system of the corresponding Cartan matrix and some special null roots.

正定矩阵与对合：它们的无穷维类似物的情形

Positive definite matrices and involutions: the manners of their infinite cousins

研究\(B(\mathbb{R})\)和\(B(\mathbb{C})\)中矩阵与对合，证明正定对合由正定矩阵共轭相应对合给出，正定矩阵有乔列斯基分解，还构造例子说明其在列向量作用下不一定有特征值及在\(B(\mathbb{C})\)中不一定有平方根。

On positive Matrices which have a Positive Smith Normal Form

It is known that any symmetric matrix $M$ with entries in $\R[x]$ and which is positive semi-definite for any substitution of $x\in\R$, has a Smith normal form whose diagonal coefficients are constant sign polynomials in $\R[x]$. We generalize this result by considering a symmetric matrix $M$ with entries in a formally real principal domain $A$, we assume that $M$ is positive semi-definite for any ordering on $A$ and, under one additionnal hypothesis concerning non-real primes, we show that the Smith normal of $M$ is positive, up to association. Counterexamples are given when this last hypothe…