Geodesic
New results about semi-positive matrices
Czechoslovak Mathematical Journal, Tome 66 (2016) no. 3, pp. 621-632
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Our purpose is to present a number of new facts about the structure of semipositive matrices, involving patterns, spectra and Jordon form, sums and products, and matrix equivalence, etc. Techniques used to obtain the results may be of independent interest. Examples include: any matrix with at least two columns is a sum, and any matrix with at least two rows, a product, of semipositive matrices. Any spectrum of a real matrix with at least $2$ elements is the spectrum of a square semipositive matrix, and any real matrix, except for a negative scalar matrix, is similar to a semipositive matrix. M-matrices are generalized to the non-square case and sign patterns that require semipositivity are characterized.
Our purpose is to present a number of new facts about the structure of semipositive matrices, involving patterns, spectra and Jordon form, sums and products, and matrix equivalence, etc. Techniques used to obtain the results may be of independent interest. Examples include: any matrix with at least two columns is a sum, and any matrix with at least two rows, a product, of semipositive matrices. Any spectrum of a real matrix with at least $2$ elements is the spectrum of a square semipositive matrix, and any real matrix, except for a negative scalar matrix, is similar to a semipositive matrix. M-matrices are generalized to the non-square case and sign patterns that require semipositivity are characterized.
DOI : 10.1007/s10587-016-0282-x
Classification : 15A23, 15A39, 15A86, 15B48, 15B52
Keywords: sign semipositivity; semipositive matrix; M-matrix; spectrum; equivalence 
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Dorsey, Jonathan; Gannon, Tom; Johnson, Charles R.; Turnansky, Morrison. New results about semi-positive matrices. Czechoslovak Mathematical Journal, Tome 66 (2016) no. 3, pp. 621-632. doi: 10.1007/s10587-016-0282-x

[1] Berman, A., Neuman, M., Stern, R. J.: Nonnegative Matrices in Dynamic Systems. Pure and Applied Mathematics, A Wiley Interscience Publication John Wiley and Sons, New York (1989). | MR

[2] Berman, A., Plemmons, R. J.: Nonnegative Matrices in the Mathematical Sciences. SIAM, Philadelphia Classics in Applied Mathematics (1994). | MR | Zbl

[3] Berman, A., Plemmons, R. J.: Nonnegative Matrices in the Mathematical Sciences. Computer Science and Applied Mathematics Academic Press, New York (1979). | MR | Zbl

[4] Berman, A., Ward, R. C.: Classes of stable and semipositive matrices. Linear Algebra Appl. 21 (1978), 163-174. | DOI | MR | Zbl

[5] Berman, A., Ward, R. C.: Stability and semipositivity of real matrices. Bull. Am. Math. Soc. 83 (1977), 262-263. | DOI | MR | Zbl

[6] Fiedler, M., Pták, V.: Some generalizations of positive definiteness and monotonicity. Number. Math. 9 163-172 (1966). | DOI | MR | Zbl

[7] Gale, D.: The Theory of Linear Economic Models. McGraw-Hill Book, New York (1960). | MR

[8] Horn, R., Johnson, C. R.: Topics in Matrix Analysis. Cambridge University Press, Cambridge (1991). | MR | Zbl

[9] Horn, R. A., Johnson, C. R.: Matrix Analysis. Cambridge University Press, Cambridge (1985). | MR | Zbl

[10] Johnson, C. R., Kerr, M. K., Stanford, D. P.: Semipositivity of matrices. Linear Multilinear Algebra 37 (1994), 265-271. | DOI | MR | Zbl

[11] Johnson, C. R., McCuaig, W. D., Stanford, D. P.: Sign patterns that allow minimal semipositivity. Linear Algebra Appl. 223/224 (1995), 363-373. | MR | Zbl

[12] Johnson, C. R., Stanford, D. P.: Qualitative semipositivity. Combinatorial and graph-theoretical problems in linear algebra IMA Vol. Math. Appl. 50 Springer, New York (1993), 99-105. | MR | Zbl

[13] Johnson, C. R., Zheng, T.: Equilibrants, semipositive matrices, calculation and scaling. Linear Algebra Appl. 434 (2011), 1638-1647. | MR | Zbl

[14] Mangasarian, O. L.: Nonlinear Programming. McGraw-Hill Book, New York (1969). | MR | Zbl

[15] Mangasarian, O. L.: Characterizations of real matrices of monotone kind. SIAM Review 10 439-441 (1968). | DOI | MR | Zbl

[16] Vandergraft, J. S.: Applications of partial orderings to the study of positive definiteness, monotonicity, and convergence of iterative methods for linear systems. SIAM J. Numer. Anal. 9 (1972), 97-104. | DOI | MR | Zbl

[17] Werner, H. J.: Characterizations of minimal semipositivity. Linear Multilinear Algebra 37 (1994), 273-278. | DOI | MR | Zbl

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