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

• 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
• 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
• Stationary iteration from a matrix splitting A=M-N Definition
• The condition number kappaₚ(A) = ||A||ₚ ||A⁻¹||ₚ of a nonsingular linear system Definition
• 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
• For a finite Galois extension, (αⱼ) is a base-field basis exactly when the matrix (σᵢαⱼ) is invertible Lemma
• Over an infinite base field, no nonzero polynomial vanishes at the conjugate tuple of every element 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
• Every finite Galois extension of an infinite field has a normal basis Theorem
• Every real or complex square matrix admits a PLU factorisation, and the first failed pivot marks the singular boundary Theorem
• Exact column-pivoted QR of a real or complex rank-r matrix has an invertible leading triangular block and zero trailing block Theorem
• Invertible matrix theorem: invertibility, full pivot rank, RREF I, trivial nullspace and unique solvability are equivalent Theorem
• Matrix perturbations in Ax = b obey a first-order condition-number bound Theorem
• Möbius transformations form a group and identify with the projective linear quotient of GL₂(C) Theorem
• On the invertible locus, Dinv(A)[H]=-A⁻¹HA⁻¹ Theorem
• Residual bounds on forward error for a nonsingular linear system Theorem
• Right-hand-side perturbations in Ax = b are amplified by the condition number Theorem
• The determinant differential is D det(A)[H]=tr(adj(A)H) at every matrix, and Jacobi's formula holds on the invertible locus Theorem

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)