Alphabeta MathHow statement and proof provenance work
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Transpose is linear and involutive, and
Statement
For matrices of the appropriate shapes and a scalar ,
and
Thus transpose is linear and involutive and reverses products.
Facts & Assumptions
Given: A field , matrices of the same shape, conformable matrices , and a scalar .
Transposition swaps the two entry indices (The transpose of a matrix).
Proof
Swapping indices in the entrywise sum and scalar product gives the two linearity identities, and swapping twice gives .
For conformable and , one has .
Commutativity in rewrites the sum in step 2.1 as , proving the product law entrywise.
Depends on
Used by
- Congruent matrices have the same rank; hence rank and nondegeneracy of a bilinear form are basis-independent Corollary
- Finite-dimensional Schwarz subspace-correction data Definition
- Full-rank lattices, covolume, and the dual lattice Definition
- Annihilators of closed subgroups of Euclidean space Example
- Continuous lattice-periodic functions are determined by their lattice Fourier coefficients Lemma
- Exact Schwarz local corrections are energy-orthogonal projections Lemma
- Level-uniform 1D Poisson smoothing and approximation bounds Lemma
- Orthogonality of the lattice characters over a fundamental domain Lemma
- Schwarz local matrices and exact solves are well defined Lemma
- A basis change by P changes the matrix of a bilinear form from A to P^TAP Theorem
- For the linear-first convention, a basis change by P sends a sesquilinear matrix A to P^TA σ(P); Hermitian forms satisfy A=σ(A)^T Theorem
- The Galerkin coarse correction is an energy-orthogonal projector Theorem
Dependency tree · two levels
9 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, Exercises 14–15 (standard reference, not scraped)