A criterion for unitary similarity of upper triangular matrices in general position

作者：Farenick Douglas; Futorny Vyacheslav; Gerasimova Tatiana G; Sergeichuk Vladimir V^*; Shvai Nadya

来源：Linear Algebra and Its Applications, 2011, 435(6): 1356-1369.

DOI：10.1016/j.laa.2011.03.021

• Unitary similarity
• Classification
• General position
• Frobenius norm

Each square complex matrix is unitarily similar to an upper triangular matrix with diagonal entries in any prescribed order. Let A = [a(ij)] and B = [b(ij)] be upper triangular n x n matrices that
are not similar to direct sums of square matrices of smaller sizes, or
are in general position and have the same main diagonal.
We prove that A and B are unitarily similar if and only if
parallel to h(A(k))parallel to = parallel to h(B(k))parallel to for all h is an element of C vertical bar x vertical bar and k = 1, ..., n,
where A(k) := [a(ij)](i.j=1)(k) and B(k) := [b(ij)](i.j=1)(k) are the leading principal k x k submatrices of A and B, and parallel to . parallel to is the Frobenius norm.

• 出版日期2011-9-15