Finite rectangular matrices over a commutative ring, their entries, rows and columns

Depends on

• Commutative ring
• A function is a relation $f$ with $(a,b) \in f$ and $(a,c) \in f$ implying $b = c$; $f : A \to B$, the value $f(a)$, domain and codomain
• The natural numbers $\mathbb{N}$ (von Neumann)

Used by

• The complex Jacobian determinant of a composite of equidimensional holomorphic maps is the product Corollary
• An oriented incidence matrix of a finite simple graph Definition
• Column-multilinear, alternating, normalized and antisymmetric functions on square matrices over a commutative ring Definition
• Deleted-row-and-column minors, cofactors, the cofactor matrix and the adjugate over a commutative ring Definition
• Entrywise ring-matrix operations, rectangular matrix products, identity matrices and transpose Definition
• Finite weighted directed multigraphs, weighted walks and their transfer matrices Definition
• For A∈ Mₙ(R), the coordinate endomorphism T_A:Rⁿ→ Rⁿ, with det(T_A):=det(A) and adj(T_A):=T_adj(A) Definition
• For n≥1, the determinant over a commutative ring by the Leibniz formula, and |det A| for a real matrix Definition
• Holomorphic maps ℂᵐ → ℂⁿ and the complex Jacobian matrix Definition
• Row swaps, arbitrary row scalings and row additions over a commutative ring, with reversible elementary cases distinguished Definition
• Submatrices and minors of a rectangular matrix Definition
• The adjacency matrix of a finite simple graph Definition
• The companion matrix of a monic polynomial Definition
• The row-shift companion matrix of a linear recurrence Definition
• The trace of a square matrix over a commutative ring Definition
• Upper triangular, lower triangular and diagonal square matrices over a commutative ring Definition
• Determinant is additive in one selected column but not under simultaneous whole-matrix addition Example
• For n≥ 1, determinant is a natural transformation det:GLₙ(-)⟹(-)^× from commutative rings to groups Example
• For a finite Galois extension, (αⱼ) is a base-field basis exactly when the matrix (σᵢαⱼ) is invertible Lemma
• For a field, the ring-matrix operations, invertibility and similarity agree exactly with the established field-matrix interface Proposition
• det(lvertM(Aᵢ,Eⱼ)|)_i,j=∑_π∈ Sᵣsgn(π)·#{non-intersecting π-systems} Theorem
• The Binet-Cauchy formula Theorem
• The composite of holomorphic maps is holomorphic and its complex Jacobian is the product Theorem

Dependency tree · two levels

15 results within two dependency steps of this one, each drawn at its shortest distance from it. An arrow runs from a result to what uses it, so the chart reads left to right and ends at this result, which carries a heavier outline. Every node is a link to that result. Click elsewhere on the chart to enlarge it.

Sources

• S. New, MATH 146 Linear Algebra 1 Lecture Notes, Notation 4.16 (standard reference, not scraped)