某些矩陣的高-吳數

Abstract

摘要

對於一個 n 乘 n 的矩陣 A，令 k(A) 表示數值域邊界上的點 ⟨Ax_j,├ x_j ⟩┤ 所對應的正交單位向量 x_j 的最大個數。我們稱這個數 k(A) 為高-吳數。若 A 為正規或二次矩陣，則其高-吳數 k(A) 可以明確地被計算出來。而對於一個矩陣 A 形如 B⊕C，我們證明了高-吳數為2時，其充分且必要條件為其中一個矩陣，稱之為 B，的數值域，完全落在另外一個矩陣 C 的數值域的內部且 k(C) 為2。對於一個不可約的矩陣 A，我們可以確切地決定何時其高-吳數等於n 。這些結果以及已知的 4 乘 4 矩陣的數值域的圖形，可用以決定任何一個 4 乘 4可約矩陣的高-吳數。

此外，設 A 為一個 n 乘 n (n大於或等於2) 的非負矩陣，其形如下 [■(0&A_1&&0@&0&⋱&@&&⋱&A_(m-1)@0&&&0)], 此處 m 大於或等於 3 並且對角線上所出現的零均為零方陣。若 A的實部為不可約的矩陣，則其高-吳數 k(A) 的上界為 m-1。再者，我們也得到這種矩陣的高-吳數達到其上界的充分且必要條件。除此之外，我們也研究了另外一類型的非負矩陣，稱之為雙隨機矩陣。我們證明了任何一個 3 乘 3 的雙隨機矩陣的高吳數必定為 3。另外，我們也決定了 4 乘 4 的雙隨機矩陣的數值域及其高-吳數。最後我們也考慮一般的 n 乘 n (n大於或等於5) 雙隨機矩陣，藉由其可能的數值域的圖形得到其高-吳數的下界。

ABSTRACT

For any n-"by" -n matrix" " A, let k(A) stand for the maximal number of orthonormal vectors x_j such that the scalar products ⟨Ax_j,├ x_j ⟩┤ lie in the boundary of the numerical range W(A). This number k(A) is called the Gau-Wu number of the matrix A. If A is a normal or a quadratic matrix, then the exact value of k(A) can be computed. For a matrix A of the form B⊕C, we show that k(A)=2 if and only if the numerical range of one summand, say, B, is contained in the interior of the numerical range of the other summand C and k(C)=2. For an irreducible matrix A, we can determine exactly when the value of k(A) equals the size of A. These are then applied to determine k(A) for a reducible matrix A of size 4 in terms of the shape of W(A).

Moreover, if A is an n-"by" -n (n ≥2) nonnegative matrix of the form [■(0&A_1&&0@&0&⋱&@&&⋱&A_(m-1)@0&&&0)], where m ≥3 and the diagonal zeros are zero square matrices, with irreducible real part, then k(A) has an upper bound m-1. In addition, we also obtain necessary and sufficient conditions for k(A)=m-1 for such a matrix A. The other class of nonnegative matrices we study is the doubly stochastic ones. We prove that the value of k(A) is equal to 3 for any 3-by-3 doubly stochastic matrix A. Next, for any 4-by-4 doubly stochastic matrix, we also determine its numerical range. This result can be applied to find the value of k(A) for any doubly stochastic matrix A of size 4 in terms of the shape of W(A). Furthermore, the lower bound of k(A) is also found for a general n-"by" -n (n ≥5) doubly stochastic matrix A via the possible shapes of W(A).