Defect of a unitary matrix
Defect of a unitary matrix
复制标题
酉矩阵的缺陷
DOI：
10.1016/j.laa.2008.02.036
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发表时间：
2007
影响因子：
1.1
通讯作者：
K. Życzkowski
中科院分区：
文献类型：
--
作者：
W. Tadej;K. Życzkowski
我们分析了一个映射f发送一个酉矩阵U的大小为N到一个双随机矩阵B=f（U）定义的Bi，j = 2的性质。对于任何U，我们定义它的亏度，由在对应于f的切映射Df下与U处的酉矩阵U的流形相切的空间TUU的像Df（TUU）的维数确定。一般酉矩阵的亏度U等于零，给出了由不等价酉矩阵U映射到同一双随机矩阵B=f（U）的光滑轨道（流形）维数的上界.我们证明了缺陷的几个性质，并证明了一个明确的公式的缺陷的傅立叶矩阵FN的大小为N。这样，我们得到了一个上界的维数的光滑轨道的不等价酉复Hadamard矩阵源于FN。它等于零当且仅当N是素数，并且如果N是素数的幂，则它与已知轨道的维数一致。这些轨道的两个结构在这项工作的最后。
We analyze properties of a map f sending a unitary matrix U of size N into a doubly stochastic matrix B=f(U) defined by Bi,j=∣Ui,j∣2. For any U we define its defect, determined by the dimension of the image Df(TUU) of the space TUU tangent to the manifold of unitary matrices U at U under the tangent map Df corresponding to f. The defect of U equal to zero for a generic unitary matrix, gives an upper bound for the dimension of a smooth orbit (a manifold) stemming from U of inequivalent unitary matrices mapped into the same doubly stochastic matrix B=f(U). We demonstrate several properties of the defect and prove an explicit formula for the defect of the Fourier matrix FNof size N. In this way we obtain an upper bound for the dimension of a smooth orbit of inequivalent unitary complex Hadamard matrices stemming from FN. It is equal to zero iff N is prime and coincides with the dimension of the known orbits if N is a power of a prime. Two constructions of these orbits are presented at the end of this work.