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A combining method and construction of order 4 case in nonnegative inverse eigenvalue problems.

  • 황진옥 (Jin Ok Hwang, 대학원 수학과 대수학전공)

초록/요약

The nonnegative inverse eigenvalue problem(NIEP) is the problem of finding necessary and sufficient conditions for a list of complex numbers to be the spectrum of some nonnegative matrix. The nonnegative inverse eigenvalue problem with complex numbers has been solved only for n=3 by Loewy and London [10]. The case n=4 and n=5 have been solved for matrices of trace zero by Reams [14] and Laffey and Meehan [18], respectively. In this paper, the first aim we consider is to solve a problem in the nonnegative inverse eigenvalue problem, that is proposed by Thomas J. Laffey and Helena Smigoc [25]. We will call this problem Laffey-Smigoc problem. As corollaries, we have new sufficient conditions of NIEP of order higher than 3. The second aim is to solve the nonnegative inverse eigenvalue problem of order 4 with complex numbers.

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목차

1. Introduction. 3
2. Preliminary. 7
3. A combining method of two nonnegative matrices which preserves the eigenvalues. 12
3.1. Preliminary-Principal submatrices for perturbation. 12
3.2. The main result for a combining method. 14
3.3. Analysis for the case of order 4. 16
3.3.1. Analysis of order 4 with t=2. 16
3.3.2. Example with t=2. 20
3.3.3. Analysis of order 4 with t=1. 22
3.3.4. An example with t=1. 24
3.3.5. A result of order 4 for t=1 and t=2. 25
3.4. Analysis for the case of order 5. 26
3.4.1. Analysis of order 5 with t=1. 26
3.4.2. An example with t=1. 28
3.4.3. Analysis of order 5 with t=2. 29
3.5. Examples of large orders with t>=2. 32
4. The nonnegative inverse eigenvalu problem of order 4 with complex numbers. 35
4.1. Preliminary. 36
4.2. The necessary conditions of order 4. 39
4.3. The nonneghative inverse eigenvalue problem with order 4. 40
4.3.1. The realizable subregions of D_N. 40
4.3.2. The realizable subregions of D_P. 45
4.3.3. The non-realizable subregions of D_P. 51

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