Entrywise ring-matrix operations, rectangular matrix products, identity matrices and transpose

Depends on

• Finite rectangular matrices over a commutative ring, their entries, rows and columns
• A finite sum in a commutative monoid indexed by an arbitrary finite set
• Commutative ring

Used by

• The determinant is alternating and multilinear in the rows as well as in the columns Corollary
• Deleted-row-and-column minors, cofactors, the cofactor matrix and the adjugate over a commutative ring Definition
• For A∈ Mₙ(F), the characteristic polynomial is χ_A(x)=det(xIₙ-A) when n≥1, with χ_A(x)=1 for the unique 0×0 matrix 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
• Invertible square matrices and similarity over a commutative ring Definition
• Matrix equivalence and Smith normal form over a PID Definition
• For n≥ 1, determinant is a natural transformation det:GLₙ(-)⟹(-)^× from commutative rings to groups Example
• False statement: every right Noetherian ring is left Noetherian False statement
• FALSE: det(A+B)=det(A)+det(B) for all same-sized square matrices False statement
• FALSE: substituting a fixed matrix A for x defines a ring homomorphism Mₙ(F[x])→ Mₙ(F) False statement
• For a finite Galois extension, (αⱼ) is a base-field basis exactly when the matrix (σᵢαⱼ) is invertible Lemma
• For A∈ Mₙ(R) and columns u,v over a commutative ring, det(A+uv^T)=det(A)+v^Tadj(A)u Lemma
• Formally, (I-xA)⁻¹=∑_n≥0Aⁿ xⁿ over every commutative coefficient ring Lemma
• Over an infinite base field, no nonzero polynomial vanishes at the conjugate tuple of every element Lemma
• The center of Mₙ(k) consists of the scalar matrices Lemma
• The quaternion double cover generates the third homotopy group of SO(3) Lemma
• The real Jacobian determinant of a complex-linear automorphism is the squared modulus of its complex determinant Lemma
• For a field, the ring-matrix operations, invertibility and similarity agree exactly with the established field-matrix interface Proposition
• The companion matrix advances the recurrence state vector by one step Proposition
• Cramer's rule over a commutative ring: every solution satisfies det(A)xⱼ=det(Aⱼ(b)), and a unit determinant gives the unique quotient formula Theorem
• Every finite Galois extension of an infinite field has a normal basis Theorem
• Every nonzero commutative ring has invariant basis number for finite bases Theorem
• For every positive-sized square matrix over a commutative ring, Aadj(A)=adj(A)A=det(A)I Theorem
• For every square matrix over a commutative ring, det(A^T)=det(A) Theorem
• Matrix arithmetic over a commutative ring is associative, unital and distributive, and transpose reverses products Theorem
• Over a commutative ring, adj(P⁻¹AP)=P⁻¹adj(A)P for every invertible P Theorem
• The (u,v) entry of Aⁿ is the total weight of length-n walks from u to v Theorem
• The Binet-Cauchy formula Theorem

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. New, MATH 146 Linear Algebra 1 Lecture Notes, Ch. 4 (standard reference, not scraped)
• P. Massot, Structures algébriques fondamentales, §6.4 (standard reference, not scraped)