Diagonals and numerical ranges of direct sums of matrices |
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Authors: | Hsin-Yi Lee |
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Affiliation: | Department of Applied Mathematics, National Chiao Tung University, Hsinchu 30010, Taiwan, ROC |
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Abstract: | For any n-by-n matrix A , we consider the maximum number k=k(A) for which there is a k-by-k compression of A with all its diagonal entries in the boundary ∂W(A) of the numerical range W(A) of 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). |
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Keywords: | 15A60 |
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