Matrix Structures & Matroids

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Abstractions about structured matrices and their combinatorial abstractions, covering sparse and patterned matrix classes (anti-diagonal, Hessenberg, pentadiagonal, and skyline matrices), specialized determinant and matrix conditions (Moore determinant, G-matrix), and matroid structures that generalize linear and algebraic independence.

10 abstractions in this family — domain-specific abstractions that sit near one another in structural-signature space (k-means over structural-signature embeddings). Each is shown with its short description.

• Algebraic Matroid — In mathematics, an algebraic matroid is a matroid, a combinatorial structure, that expresses an abstraction of the relation of algebraic independence.
• Anti-Diagonal Matrix — In mathematics, an anti-diagonal matrix is a square matrix where all the entries are zero except those on the diagonal going from the lower left corner to the upper right corner (↗) , known as the anti-diagonal (sometimes Harrison diagonal, secondary diagonal, trailing diagonal, minor diagonal, off diagonal or bad diagonal).
• Dual matroid — In matroid theory, the dual of a matroid M is another matroid M^\ast that has the same elements as M , and in which a set is independent if and only if M has a basis set disjoint from it.
• G-Matrix — In linear algebra, a real invertible matrix A is called a G-matrix if A^{-T}=D_1AD_2 (where A^{-T} means (A^{-1})T ) for some real diagonal matrices D_1 and D_2 .
• Hessenberg Matrix — Constrain a square matrix to be triangular except for one adjacent off-diagonal band, yielding a similarity-reachable form that preserves eigenvalues while making QR iteration and related computations substantially cheaper.
• Integer matrix — In mathematics, an integer matrix is a matrix whose entries are all integers.
• LU Decomposition — A square-matrix factorization into lower- and upper-triangular factors, with permutations when needed.
• Moore determinant of a Hermitian matrix — In mathematics, the Moore determinant is a determinant defined for Hermitian matrices over a quaternion algebra, introduced by .
• Pentadiagonal Matrix — In mathematics, particularly matrix theory, a band matrix or banded matrix is a sparse matrix whose non-zero entries are confined to a diagonal band, comprising the main diagonal and zero or more diagonals on either side.
• Skyline matrix — In scientific computing, skyline matrix storage, or SKS, or a variable band matrix storage, or envelope storage scheme is a form of a sparse matrix storage format matrix that reduces the storage requirement of a matrix more than banded storage.