Combinatorial matrix theory

Combinatorial matrix theory is a branch of linear algebra and combinatorics that studies matrices in terms of the patterns of nonzeros and of positive and negative values in their coefficients.[1][2][3]

Concepts and topics studied within combinatorial matrix theory include:

  • (0,1)-matrix, a matrix whose coefficients are all 0 or 1
  • Permutation matrix, a (0,1)-matrix with exactly one nonzero in each row and each column
  • The Gale–Ryser theorem, on the existence of (0,1)-matrices with given row and column sums
  • Hadamard matrix, a square matrix of 1 and –1 coefficients with each pair of rows having matching coefficients in exactly half of their columns
  • Alternating sign matrix, a matrix of 0, 1, and –1 coefficients with the nonzeros in each row or column alternating between 1 and –1 and summing to 1
  • Sparse matrix, a matrix with few nonzero elements, and sparse matrices of special form such as diagonal matrices and band matrices
  • Sylvester's law of inertia, on the invariance of the number of negative diagonal elements of a matrix under changes of basis

Researchers in combinatorial matrix theory include Richard A. Brualdi and Pauline van den Driessche.

References
  1. Brualdi, Richard A.; Ryser, Herbert J. (1991), Combinatorial matrix theory, Encyclopedia of Mathematics and its Applications, 39, Cambridge University Press, Cambridge, doi:10.1017/CBO9781107325708, ISBN 0-521-32265-0, MR 1130611
  2. Brualdi, Richard A. (2006), Combinatorial matrix classes, Encyclopedia of Mathematics and its Applications, 108, Cambridge University Press, Cambridge, doi:10.1017/CBO9780511721182, ISBN 978-0-521-86565-4, MR 2266203
  3. Brualdi, Richard A.; Carmona, Ángeles; van den Driessche, P.; Kirkland, Stephen; Stevanović, Dragan (2018), Combinatorial matrix theory: Notes of the lectures delivered at Centre de Recerca Matemàtica (CRM), Bellaterra, June 29–July 3, 2015, Advanced Courses in Mathematics. CRM Barcelona, Birkhäuser/Springer, Cham, p. xi+217, doi:10.1007/978-3-319-70953-6, ISBN 978-3-319-70952-9, MR 3791450
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