On unitary equivalence of arbitrary matrices
On unitary equivalence of arbitrary matrices
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DOI：
10.1090/s0002-9947-1962-0140525-0
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发表时间：
1962-02
影响因子：
1.3
通讯作者：
H. Radjavi
中科院分区：
文献类型：
--
作者：
H. Radjavi
1. 简介。我们希望研究的问题是确定复数域上的两个给定方阵 A 和 B 是否酉等价，即是否存在酉矩阵 U 使得 B = U-'A U。如果获得所有矩阵的一组可计算的规范形式，则可以轻松做出此决定，即是否存在与任何给定矩阵 A 关联到另一个矩阵 C(A) 的算法，使得如果 A 和 B 是两个矩阵且 C(A)和 C(B) 分别由算法得到的形式，则 C(A) 等于 C(B) 当且仅当 A 和 B 酉等价。该问题对于正规矩阵集的解决方案是众所周知的；规范集由所有对角矩阵组成，其中复杂的条目按商定的某种顺序排列。我们将利用有关正规矩阵对角化的事实。当前问题的类似情况，即考虑相似性而不是单一等价性，要简单得多（乔丹规范形式）。在酉等价的情况下，我们不能期望有一个简单的规范集。下面的示例显示了这种情况下的规范形式集与 Jordan 规范形式集相比可以大多少： 令 n> 2。取以下形式的所有 n X n 矩阵
1. Introduction. The problem we wish to study is that of deciding whether two given square matrices A and B over the field of complex numbers are unitarily equivalent, i.e., whether there exists a unitary matrix U such that B = U-'A U. This decision can be made easily if a computable set of canonical forms for all matrices is obtained, that is, if there exists an algorithm which associates with any given matrix A another matrix C(A) such that if A and B are two matrices and C(A) and C(B) their respective forms obtained by the algorithm, then C(A) is equal to C(B) if and only if A and B are unitarily equivalent. The solution of this problem for the set of normal matrices is well known; the canonical set consists of all diagonal matrices with complex entries arranged in some order agreed on. We shall make use of facts concerning the diagonalization of normal matrices. The analog of the present problem, where similarity is considered instead of unitary equivalence is much simpler (Jordan canonical forms). We cannot expect as simple a canonical set in the case of unitary equivalence. The following example shows how much vaster the set of canonical forms in this case can be as compared to the set of Jordan canonical forms: Let n> 2. Take all n X n matrices of the form