Defect of a unitary matrix

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• 作者：Wojciech Tadej ; Karol Ż ; yczkowski
• 关键词：Unitary matrices ; Bistochastic matrices ; Critical point ; Fourier matrices ; Complex Hadamard matrices
• 刊名：Linear Algebra and its Applications
• 出版年：2008
• 期刊代码：38_00243795
• 类别：mt
• 出版时间：15 July 2008
• 卷：429
• 期：2-3
• 页码：447-481
• 文件大小：342 K

摘要

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 B_i,j = der="0">U_i,jder="0">². For any U we define its defect, determined by the dimension of the image SBHFKC-1&_mathId=mml1&_user=1067359&_cdi=5653&_rdoc=4&_acct=C000050221&_version=1&_userid=10&md5=aafd2f9bb4b7b74f3d19d45f9413008e">sbottom" border="0" height=17 width="69"/> of the space sbottom" border="0" height=16 width="36"/> tangent to the manifold of unitary matrices de9ecd8b6eecfe35cb713">sbottom" border="0" height=13 width="15"/> 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 F_N of 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 F_N. 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.