Hermitian matrices over polynomial rings
Hermitian matrices over polynomial rings
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多项式环上的埃尔米特矩阵
DOI：
10.1016/0021-8693(76)90119-8
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发表时间：
1976
影响因子：
0.9
通讯作者：
D. Djoković
中科院分区：
文献类型：
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作者：
D. Djoković
设R是扭多项式环F [X; S，D]，其中F是除环，S是F的自同构，D是F的S-导子.因此Xα= α S X+ α D对每个α <$F成立。设F是R的一个对合，使得F = F。然后证明了R上的各向异性厄米特矩阵a与矩阵B全等，使得B的对角元素的次数严格大于同一行中任意非对角元素的次数.由此可见，如果a也是幺模的，那么它一定与一个对角矩阵全等。若F= CorH（复形或四元数）且S= 1，D= 0，X = X，而F的限制条件是共轭，则R上的每个半正定矩阵a都有一个分解a= B <$B，其中B是R上的方阵。
Let R be a twisted polynomial ring F [X; S, D] where F is a division ring, S is an automorphism of F and D is an S-derivation of F. Thus Xα= α S X+ α D holds for every α ϵ F. Let∗ be an involution of R such that F∗= F. Then we show that every anisotropic hermitian matrix a over R is congruent to a matrix b such that the degree of every diagonal entry of b is strictly larger than the degree of any off-diagonal entry in the same row. It follows that if a is also unimodular then it must be congruent to a diagonal matrix. If F= CorH (complexes or quaternions) and S= 1, D= 0, X∗= X while the restriction of∗ to F is the conjugation then every positive semidefinite matrix a over R admits a factorization a= b∗ b with b a square matrix over R.