2 × 2和3 × 3矩阵在扰动下的合同标准形的变化及合同矩阵丛的变化
Change of the congruence canonical form of 2-by-2 and 3-by-3 matrices under perturbations and bundles of matrices under congruence
摘要：
我们构造了$2\times 2$和$3\times 3$复矩阵的同余类集合上的闭包序的Hasse图$G_2$和$G_3$.换句话说，我们构造了两个有向图，其顶点分别是$2\times 2$或$3\times 3$典范矩阵下的同余，并且存在从$A$到$B$的有向路径当且仅当$A$可以通过任意小的扰动变换为与$B$全等的矩阵。同余下矩阵丛定义为方阵$A$的集合，在严格等价下，其束$A+λA^T$属于同一丛。为了支持这个定义，我们证明了$2\times 2$或$3\times 3$矩阵的合同丛中的所有矩阵对于扰动具有相同的性质。我们分别构造了$2\times 2$和$3\times 3$矩阵的同余丛集合上的闭包序的Hasse图$G_2^{\rm B}$和$G_3^{\rm B}$.我们找到了$2\times 2$和$3\times 3$同余典范矩阵的等距群.
英文摘要：
We construct the Hasse diagrams $G_2$ and $G_3$ for the closure ordering on the sets of congruence classes of $2\times 2$ and $3\times 3$ complex matrices. In other words, we construct two directed graphs whose vertices are $2\times 2$ or, respectively, $3\times 3$ canonical matrices under congruence and there is a directed path from $A$ to $B$ if and only if $A$ can be transformed by an arbitrarily small perturbation to a matrix that is congruent to $B$. A bundle of matrices under congruence is defined as a set of square matrices $A$ for which the pencils $A+λA^T$ belong to the same bundle under strict equivalence. In support of this definition, we show that all matrices in a congruence bundle of $2\times 2$ or $3\times 3$ matrices have the same properties with respect to perturbations. We construct the Hasse diagrams $G_2^{\rm B}$ and $G_3^{\rm B}$ for the closure ordering on the sets of congruence bundles of $2\times 2$ and, respectively, $3\times 3$ matrices. We find the isometry groups of $2\times 2$ and $3\times 3$ congruence canonical matrices.
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