Rectangular matrix multiplication and the identity matrix $I_n$, including zero-sized shapes

Depends on

• The vector space $M_{m \times n}(F) := F^{\,m \times n}$ of $m$ by $n$ matrices over a field, with entrywise operations
• A finite sum in a commutative monoid indexed by an arbitrary finite set

Used by

• The rank of a matrix equals the rank of the linear map x↦ Ax Corollary
• BCH truncation fails when higher commutators do not vanish Counterexample
• A finite family of subsets of [n] and its incidence matrix over F Definition
• A normalised LU factorisation has a unit lower-triangular L and an upper-triangular U Definition
• Block LU factorisation and the associated Schur complement Definition
• Elementary matrices obtained by applying one elementary row operation to an identity matrix Definition
• Forward and backward substitution for triangular linear systems Definition
• Invertible matrices and the general linear group GLₙ(F) Definition
• Matrix equation Ax=b, its solution set, consistency, homogeneous systems and the augmented matrix [A∣ b] Definition
• Permutation matrices, partial pivoting, and the pivot-growth factor Definition
• Stationary iteration from a matrix splitting A=M-N Definition
• The GMRES iterate as the residual minimizer over an affine Krylov space Definition
• The inclusion matrix W_t,k(n) of t-sets against k-sets Definition
• The Krylov subspace Kₘ(A,b)=span{b,Ab,…,Aᵐ⁻¹b} Definition
• The matrix norm induced by a published vector p-norm Definition
• The matrix-free access model for Krylov methods Definition
• The Moore--Penrose pseudoinverse A^+ as the solution of the four Penrose equations Definition
• The mth Krylov Galerkin iterate in the affine space x₀+Kₘ(A,r₀) for a Hermitian positive-definite system Definition
• The reduced resolvent, or group inverse, on the complementary invariant subspace of a simple eigenvalue Definition
• The residual r=Ax-μ x and the normwise backward error of an approximate eigenpair Definition
• Chosen bases exhibit Mat_F as equivalent to finite-dimensional vector spaces Example
• Curvature two-form of a connection on a trivial plane bundle Example
• Matrix exponential as the Lie-group exponential Example
• W_1,2(4) written out, and its rank computed Example
• The exponential map is surjective on every connected Lie group False statement
• The exponential map of a Lie group is a group homomorphism False statement
• A coordinate scaling and a coordinate transposition send the unit cube to a set of measure equal to the absolute value of the determinant Lemma
• A shear sends the unit cube to a set of Lebesgue measure one Lemma
• Determinant criterion: if the matrix (⟨ vᵢ,wⱼ⟩)_i,j is invertible then both v₁,…,vₘ and w₁,…,wₘ are linearly independent Lemma
• EᵢⱼE_kℓ=δⱼₖE_iℓ Lemma
• For a field, the ring-matrix operations, invertibility and similarity agree exactly with the established field-matrix interface Proposition
• Matrix differentials obey the sum rule, product rule, and adjoint rule Proposition
• Transpose is linear and involutive, and (AB)^T=B^TA^T Proposition
• [T(v)]_C=[T]_B^C[v]_B Theorem
• A linear map T of ℝⁿ sends Lebesgue measurable sets to Lebesgue measurable sets, with λₙ(T[E])=|det T| λₙ(E) when T is invertible and T[E] Lebesgue null when it is not Theorem
• A matrix admits a Cholesky factorisation with positive diagonal exactly when it is Hermitian positive definite, and that factor is unique Theorem
• An invertible leading block yields block LU through its Schur complement Theorem
• Applying an elementary row operation is left multiplication by its elementary matrix Theorem
• For A∈ M_m× n(F) and B∈ M_n× m(F), tr(AB)=tr(BA) Theorem
• Freivalds' one-trial matrix-product test has one-sided error Theorem

…and 4 more results.

Dependency tree · two levels

13 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. Axler, Linear Algebra Done Right, 4th ed., §3C (standard reference, not scraped)