论文标题
h-非态矩阵的平方根
Square roots of H-nonnegative matrices
论文作者
论文摘要
矩阵的根是经过充分研究的。它们存在的条件是理解的：成对排列的Nilpotent Jordan块的块大小必须满足一些简单的代数属性。 结构化矩阵的结构化根更有趣。最著名的例子可能是正确定矩阵的正定正方根的存在和独特性。如果一个人掉下了平方根正确定性的要求，则事实证明存在丰富的平方根。在这里，所有平方根的所有规范形式都可能是可能的，并且是直截了当的。 H-非能型矩阵是H-selfadjoint，对于与Gramian H的无限期内部产品无关。h-nonenegative矩阵$ b $允许分解负面的确定性，nilpotent h-nonnenegative，n-nonnonnenegative，b = b_-f = b_-f = b = b = b = b = b = b = b_- \ oplus b_0 b_ b_ b_ b_+oplus b_+oplus b_+oplus b_+oplus。有趣的部分是B_0，因为只有Jordan大小的Jordan块发生。确定B的平方根还原以确定B_-，B_0和B_+的每个平方根。在这里，我们调查了H-非能基质：它的平方根没有其他结构，以及其Honnengative或H-Selfadhixhixhint的结构化方形根。 对于这三类H非核矩阵的平方根，我们显示了它们存在的简单标准，并描述了所有可能的规范形式。这主要基于已知结果，但一个重要的新部分是，在所有三种情况下，我们描述了nilpotent H-非元件矩阵B_0的所有可能的平方根。此外，我们展示了如何将结果应用于H-非态矩阵H-非核方形根的条件和无条件稳定性，其中使用了B_0的平方根的明确描述。
Roots of matrices are well-studied. The conditions for their existence are understood: The block sizes of nilpotent Jordan blocks, arranged in pairs, have to satisfy some simple algebraic property. More interesting are structured roots of structured matrices. Probably the best known example is the existence and uniqueness of positive definite square roots of a positive definite matrix. If one drops the requirement of positive definiteness of the square root, it turns out that there exists an abundance of square roots. Here a description of all canonical forms of all square roots is possible and is straight forward. H-nonnegative matrices are H-selfadjoint and are nonnegative with respect to an indefinite inner product with Gramian H. An H-nonnegative matrix $B$ allows a decomposition in a negative definite, a nilpotent H-nonnegative, and a positive definite matrix, B=B_- \oplus B_0 \oplus B_+. The interesting part is B_0, as only Jordan blocks of size one and two occur. Determining a square root of B reduces to determining a square root of each of B_-, B_0, and B_+. Here we investigate for an H-nonnegative matrix: its square roots without additional structure, as well as its structured square roots that are H-nonnegative or H-selfadjoint. For these three classes of square roots of H-nonnegative matrices we show a simple criterion for their existence and describe all possible canonical forms. This is based mainly on known results but an important new part is that in all three cases we describe all possible square roots of the nilpotent H-nonnegative matrix B_0 explicitly. Moreover, we show how our results can be applied to the conditional and unconditional stability of H-nonnegative square roots of H-nonnegative matrices, where the explicit description of the square roots of B_0 is used.
