Mappings on matrices: invariance of functional values of matrix products
Mappings on matrices: invariance of functional values of matrix products
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DOI：
10.1017/s1446788700015809
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发表时间：
2006-10
影响因子：
0.7
通讯作者：
J. Chan;Chi-Kwong Li;Nung-Sing Sze
中科院分区：
文献类型：
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作者：
J. Chan;Chi-Kwong Li;Nung-Sing Sze
设Mn是域F上所有n × n矩阵的代数，其中n ≠ 2。设S是Mn的一个包含所有秩为1的矩阵的子集。我们研究映射φ：S → Mn，使得F（φ（A）φ（B））= F（A B），对于各种函数族F，包括真实的或复矩阵上的所有酉相似不变函数。这些映射通常具有形式A <$μ（A）S（σ（aij））S-1，其中A =（aij）∈ S，S ∈ Mn，F的域单态σ，以及定义在S上的F * 值映射μ.对于实矩阵，σ通常是单位映射;对于复矩阵，σ通常是单位映射或共轭映射：z <$z。我们研究中的一个关键思想是将问题简化为F：Mn → {0，1}定义为F（X）= 0的特殊情况，如果X = 0，否则F（X）= 1。在这种情况下，需要刻画φ：S → Mn使得φ（A）φ（B）= 0当且仅当AB = 0。我们证明了这样一个映射在S中的秩一矩阵上具有上述的标准形式。
Abstract Let Mn, be the algebra of all n × n matrices over a field F, where n ≧ 2. Let S be a subset of Mn containing all rank one matrices. We study mappings φ: S → Mn, such that F(φ (A)φ (B)) = F(A B) for various families of functions F including all the unitary similarity invariant functions on real or complex matrices. Very often, these mappings have the form A ↦ μ(A)S(σ (aij))S-1 for all A= (aij) ∈ S for some invertible S ∈ Mn, field monomorphism σ of F, and an F*-valued mapping μ defined on S. For real matrices, σ is often the identity map; for complex matrices, σ is often the identity map or the conjugation map: z ↦ z. A key idea in our study is reducing the problem to the special case when F:Mn → {0, 1} is defined by F(X) = 0, if X = 0, and F(X) = 1 otherwise. In such a case, one needs to characterize φ: S → Mn such that φ(A) φ (B) = 0 if and only if AB = 0. We show that such a map has the standard form described above on rank one matrices in S.