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.{{citation

| last1 = Brualdi

| first1 = Richard A.

| last2 = Ryser

| first2 = Herbert J.

| doi = 10.1017/CBO9781107325708

| isbn = 0-521-32265-0

| mr = 1130611

| publisher = Cambridge University Press, Cambridge

| series = Encyclopedia of Mathematics and its Applications

| title = Combinatorial matrix theory

| volume = 39

| year = 1991

| url-access = registration

| url = https://archive.org/details/combinatorialmat0000brua_x9u3

}}{{citation

| last = Brualdi

| first = Richard A.

| doi = 10.1017/CBO9780511721182

| isbn = 978-0-521-86565-4

| mr = 2266203

| publisher = Cambridge University Press, Cambridge

| series = Encyclopedia of Mathematics and its Applications

| title = Combinatorial matrix classes

| volume = 108

| year = 2006

| url-access = registration

| url = https://archive.org/details/combinatorialmat0000brua

}}{{citation

| last1 = Brualdi | first1 = Richard A.

| last2 = Carmona | first2 = Ángeles

| last3 = van den Driessche | first3 = P.

| last4 = Kirkland | first4 = Stephen

| last5 = Stevanović | first5 = Dragan

| doi = 10.1007/978-3-319-70953-6

| isbn = 978-3-319-70952-9

| mr = 3791450

| page = xi+217

| publisher = Birkhäuser/Springer, Cham

| series = Advanced Courses in Mathematics. CRM Barcelona

| title = Combinatorial matrix theory: Notes of the lectures delivered at Centre de Recerca Matemàtica (CRM), Bellaterra, June 29–July 3, 2015

| year = 2018}}

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.