ㄧ個n階矩陣A其缺陷指數為$rank(I_n-A^ast A)$。本論文探討關於「缺陷指數為1的矩陣」其性質之刻劃。令$mathcal{S}_n ≡ {A in M_n: rank(I_n-A^ast A)=1 and |lambda|<1 for all lambda in sigma(A)}$和 $mathcal{S}_n^{-1} ≡ {A in M_n: rank(I_n-A^ast A)=1 and |lambda|>1 for all lambda in sigma(A)}$。首先我們發現這兩類矩陣皆為具有缺陷指數為1之基本矩陣，進一步而言，我們證明一矩陣其缺陷指數為1之充分必要條件為它可分解成一個么正矩陣和一個$mathcal{S}_n$ 矩陣的直和或是一個么正矩陣和一個$mathcal{S}_n^{-1}$矩陣的直和。此外，我們也針對$mathcal{S}_n^{-1}$矩陣給一個完整的刻劃以及它們的極分解。亦證明每一個$mathcal{S}_n^{-1}$ 矩陣均具有循環、不可分解、且其數值域之邊界為一代數曲線。並給出$mathcal{S}_n^{-1}$ 矩陣的範數其由特徵值所表示。

Given an $n$-by-$n$ matrix $A$, the dimension of $ran(I_n-A^ast A)$ is called the defect index of $A$. In this thesis, we make a detailed study of matrices $A$ with the property $rank(I_n-A^ast A)=1$. Let $mathcal{S}_n equiv {A in M_n: rank(I_n-A^ast A)=1: and |lambda|<1 for all lambda in sigma(A)}$ and $mathcal{S}_n^{-1} equiv {A in M_n: rank(I_n-A^ast A)=1 and |lambda|>1 for all lambda in sigma(A)}$. Firstly, we give a complete characterization for matrices of defect index one, namely, $rank(I_n-A^ast A)=1$ if and only if $A$ is unitarily equivalent to either $U oplus B$ or $U oplus C$, where $U$ is a $k times k$ unitary matrix, $0 leq k< n$, $B in mathcal{S}_{n-k}^{-1}$ and $C in mathcal{S}_{n-k}$. Moreover, we also give a complete characterization of $mathcal{S}_n^{-1}$-matrices. We find the polar decompositions of $mathcal{S}_n^{-1}$-matrices. Next, we prove that every $mathcal{S}_n^{-1}$-matrix is irreducible, cyclic, and the boundary of its numerical range is an algebraic curve. Finally, we give the
norm of $mathcal{S}_n^{-1}$-matrices in terms of its eigenvalues.

[1] R.A. Horn and C.R. Johnson, Matrix Analysis, Cambridge University Press, Cambridge, 1985.
[2] R.A. Horn and C.R. Johnson, Topics in Matrix Analysis, Cambridge University Press, 1991.
[3] D.S.Keeler, L. Rodman and I.M. Spitkovsky, The Numerical Range of 3×3 Matrices, Linear Algebra and its Appl., 252 (1997), 115–139.
[4] H.-L. Gau and P. Y. Wu, Numerical Range of S(ψ), Linear and Multilinear Algebra, 45 (1998), 49–73.
[5] H.-L. Gau and P. Y. Wu, Lucas’ Theorem Refined, Linear and Multilinear Algebra, 45 (1999), 359–373.
[6] H.-L. Gau and P. Y. Wu, Companion Matrices: Reducibility, Numerical Ranges and Similarity to Contractions, Linear Algebra and its Appl., 383 (2004),127–142.
[7] H.-L. Gau and P. Y. Wu, Numerical Range of a Normal Compression, Linear and Multilinear Algebra, 52 (2004), 195–201.
[8] P. Y. Wu, Polar Decompositions of C0(N) contractions, Integral Equations Operator Theory, 56 (2006), 559–569.
[9] M.-S. Sun, A Study on Reducible Companion Matrices, Master Thesis, June 2006, National Central University, Taiwan.