臺灣博碩士論文加值系統

論文名稱：位移算子其有限維壓縮算子的反矩陣
頁數：27 頁
校所組別：國立中央大學數學所甲組
研究生：張承鈞　　　　　　　指導教授：高華隆
論文提要內容：
令 A 為一個 n 階矩陣，其缺陷指數 d_A 為 rank (I_n − A∗A)。
本論文探討關於「缺陷指數為 1 的矩陣」其性質之刻劃。
令 S_n ≡ { A ∈ M_n : d_A = 1 and |λ| < 1 for all λ ∈ σ(A) } 和 S_n^−1 ≡ { A ∈ M_n : d_A = 1 and |λ| > 1 for all λ ∈ σ(A)}。我們針對「缺陷指數為 1 的矩陣」研究其極分解、數值域、範數和冪次對缺陷指數的影響。進一步而言，我們證明了 S_n^−1-矩陣其實部的特徵值皆無重根。此外，我們也對 S_n^−1-矩陣的數值域做了詳細的刻劃。最後我們給出任一矩陣為S_n−1-矩陣的充分必要條件。　

Let M_n be the algebra of all n-by-n complex matrices. Let A be an n-by-n matrix. The defect index of A is defined and denoted by d_A ≡ rank (I_n − A∗A). In this thesis, we study some unitary-equivalence properties of matrices with defect index one. We denote S_n ≡ {A ∈ M_n : d_A = 1 and |λ| < 1 for all λ ∈ σ(A)} and S_n^−1 ≡ {A ∈ M_n : d_A = 1 and |λ| > 1 for all λ ∈ σ(A)}. We want to give some characterizations of the polar decompositions, numerical ranges, norms and defect indices of powers of matrices with defect index one. In particular, we show that the eigenvalues of the real part of operators in S_n^−1 are simple. Next, we give some characterizations of the numerical ranges of S_n^−1-matrices. Finally, we find the sufficient and necessary conditions for a matrix in the class S_n^−1.

1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
2 Preliminaries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3
2.1 Basic Properties for Numerical Ranges . . . . . . . . . . . . . 3
2.2 Powers of a Contraction . . . . . . . . . . . . . . . . . . . . . . . 5
2.3 Classifications of Matrices with Defect Index One. . . . . . 6
2.4 Matrices of Classes S_n and S_n^−1 . . . . . . . . . . . . . . 7
3 Characterizations of S_n^−1-matrices . . . . . . . . . . . . . . . 10
3.1 Spectrum of Real Part . . . . . . . . . . . . . . . . . . . . . . . . 10
3.2 Numerical Ranges . . . . . . . . . . . . . . . . . . . . . . . . . . . 12
3.3 Defect Indices of Powers . . . . . . . . . . . . . . . . . . . . . . 19
Reference . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26

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