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, is 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, vol. 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, vol. 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