Invertible matrices and the general linear group ${\mathrm{GL}}_n(F)$

Depends on

• Rectangular matrix multiplication and the identity matrix $I_n$, including zero-sized shapes
• $M_n(F)$ is a ring under entrywise addition and matrix multiplication, including the zero ring $M_0(F)$

Used by

• Flexible GMRES with variable right preconditioning Algorithm
• Every elementary matrix is invertible, with inverse given by the reverse elementary operation Corollary
• GLₙ(F) is a group under matrix multiplication, including the trivial group GL₀(F) Corollary
• If A is invertible, the exact solution of Ax=b lies in the Krylov space at the grade Corollary
• Similar matrices have the same trace Corollary
• SL₂ and PGL₂ have the same Lie algebra but differ globally Counterexample
• Block LU factorisation and the associated Schur complement Definition
• Equivariant constructible derived category of complex varieties Definition
• Finite Gaussian mixture with positive-definite component covariances Definition
• Full-rank lattices, covolume, and the dual lattice Definition
• Inverse iteration and shifted inverse iteration Definition
• Left preconditioning, right preconditioning, and symmetric preconditioning of a linear system Definition
• Real and complex topological vector bundles Definition
• Similar matrices: B=P⁻¹AP for an invertible P Definition
• Standard subgroups of finite general linear groups Definition
• Stationary iteration from a matrix splitting A=M-N Definition
• Symmetric saddle-point system, Schur complement, and block preconditioners Definition
• The condition number κₚ(A) = ‖ A‖ₚ ‖ A⁻¹‖ₚ of a nonsingular linear system Definition
• The frame bundle of a smooth manifold Definition
• The modular group and its action on the upper half-plane Definition
• Annihilators of closed subgroups of Euclidean space Example
• Dual lattice and covolume for a diagonal scaling Example
• Equivariant Ext of a point is degreewise finite and not total finite Example
• Acyclicity makes the simply connected middle-handle matrix unimodular Lemma
• Continuous lattice-periodic functions are determined by their lattice Fourier coefficients Lemma
• Determinant classifies loops in complex general linear groups Lemma
• Determinant criterion: if the matrix (⟨ vᵢ,wⱼ⟩)_i,j is invertible then both v₁,…,vₘ and w₁,…,wₘ are linearly independent Lemma
• Every Whitehead class is represented by an invertible matrix and conversely Lemma
• For a finite Galois extension, (αⱼ) is a base-field basis exactly when the matrix (σᵢαⱼ) is invertible Lemma
• General linear Stiefel approximations stabilize bounded constructible Hom Lemma
• Over an infinite base field, no nonzero polynomial vanishes at the conjugate tuple of every element Lemma
• Positively oriented bases of an oriented vector space are path-connected Lemma
• Strict positive and conditionally positive augmented interpolation systems are invertible Lemma
• Triangular elimination produces a pivot permutation Lemma
• For a field, the ring-matrix operations, invertibility and similarity agree exactly with the established field-matrix interface Proposition
• If A has full column rank, then A^+=(A^*A)⁻¹A^* Proposition
• Invertible preconditioners give equivalent linear systems, with the transformed residuals and errors written explicitly Proposition
• [v]_C=P_C leftarrowB[v]_B and P_B leftarrowC=P_C leftarrowB⁻¹ Theorem
• A square matrix is invertible exactly when its multiplication map is a linear isomorphism; matrices preserve inverses of linear isomorphisms Theorem
• An invertible leading block yields block LU through its Schur complement Theorem

…and 13 more results.

Dependency tree · two levels

6 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., Definition 3.80 (standard reference, not scraped)
• S. Schiavone, MIT 18.700 Day 9, Definition 33 (standard reference, not scraped)