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Nonnegative Matrix


A nonnegative matrix is a real or integer matrix (a)_(ij) for which each matrix element is a nonnegative number, i.e., a_(ij)>=0 for all i, j.

Nonnegative matrices are therefore a superset of positive matrices.

Nonnegative matrices are important in a variety of applications and have a number of attractive mathematical properties. Together with positive semidefinite matrices, they therefore serve as a natural generalization of nonnegative real numbers (Johnson 1981). The most fundamental properties of nonnegative matrices require fairly advanced mathematics and were established by Perron (1907) and Frobenius (1912).


See also

Identity Matrix, Integer Matrix, Matrix, Negative Matrix, Nonnegative, Nonnegative Integer, Nonpositive Matrix, Positive Matrix, Real Matrix, Unit Matrix, Zero Matrix

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References

Berman, A. and Plemmons, R. J. Nonnegative Matrices in the Mathematical Sciences. SIAM, 1994.Carlson, D. Book review. "Nonnegative Matrices in the Mathematical Sciences. By Abraham Berman and Robert J. Plemmons. Academic Press, New York, 1979." SIAM Review 23, 409-410, 1981.Frobenius, G. "Über Matrizen aus nicht negativen Elementen." S.-B. Preuss. Akad. Wiss. (Berlin). pp. 456-477, 1912.Gantmacher, F. R. The Theory of Matrices, Vols. 1 and 2. New York: Chelsea, New York, 1959.Johnson, C. R. Book review. "Nonnegative Matrices in the Mathematical Sciences, by Abraham Berman and Robert J. Plemmons, Academic Press, New York, 1979." Bull. Amer. Math. Soc. 6, 233-235, 1981.Perron, O. "Zur theorie der matrizen." Math. Ann. 64, 248-263, 1907.Senta, E. Nonnegative Matrices. New York: Wiley, 1973.Taussky, O. Eigenvalues of Finite Matrices. New York: McGraw-Hill, 1962.Varga, R. S. Matrix Iterative Analysis. Englewood Cliffs, NJ: Prentice-Hall, 1962.Wielandt, H. "Unzerlegbare, nicht negative matrizen." Math. Z. 52, 642-648, 1950.

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Nonnegative Matrix

Cite this as:

Weisstein, Eric W. "Nonnegative Matrix." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/NonnegativeMatrix.html

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