论文标题
通用的pn型和符号正常矩阵的通用规范形式
Generic canonical forms for perplectic and symplectic normal matrices
论文作者
论文摘要
令$ b $成为一些可逆的赫米尔人或偏斜的矩阵。如果$ aa^\ star = a^\ star a $持有$ a $ and and a $ a $ and and usanix matrix $ a^\ star：= b^{ - 1} a^hb $，则矩阵$ a $称为$ b $ normal。此外，如果$ q^hbq = b $，矩阵$ q $被称为$ b $单位。我们为非缺陷的稀疏规范形式（即对角线化）$ j_ {2n} $ - 普通矩阵和$ r_n $ - normal矩阵下的$ j_ {2n} $ - unitary（分别$ r_n $ - 独立），$ r_n $ - 独立的相似性转换，$ j_ j_ j_ j_ j_ j_ j_ {2n} = 2n} = _ i _ i _ irix = ＆\ end {bmatrix} \ in m_ {2n}（\ mathbb {c}）$$和$ r_n $是$ n \ times n $ sip矩阵，其在其他地方的抗diagonal和Zeros上。在这两种情况下，我们都表明，这些表格存在于$ j_ {2n}/r_n $ -Normal矩阵的开放和密集子集中。这意味着这些形式可以被视为拓扑上的“通用”，对于$ j_ {2n}/r_n $ - 正常矩阵，因为它们存在于所有此类矩阵外，除了无处密集的子集。
Let $B$ be some invertible Hermitian or skew-Hermitian matrix. A matrix $A$ is called $B$-normal if $AA^\star = A^\star A$ holds for $A$ and its adjoint matrix $A^\star := B^{-1}A^HB$. In addition, a matrix $Q$ is called $B$-unitary, if $Q^HBQ = B$. We develop sparse canonical forms for nondefective (i.e. diagonalizable) $J_{2n}$-normal matrices and $R_n$-normal matrices under $J_{2n}$-unitary ($R_n$-unitary, respectively) similarity transformations where $$J_{2n} = \begin{bmatrix} & I_n \\ - I_n & \end{bmatrix} \in M_{2n}(\mathbb{C})$$ and $R_n$ is the $n \times n$ sip matrix with ones on its anti-diagonal and zeros elsewhere. For both cases we show that these forms exist for an open and dense subset of $J_{2n}/R_n$-normal matrices. This implies that these forms can be seen as topologically 'generic' for $J_{2n}/R_n$-normal matrices since they exist for all such matrices except a nowhere dense subset.
