The trace $\mathrm{tr}(A)$ as the sum of the diagonal entries

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

• Finite locally linear embedding with minimum-norm weights and a simple constant kernel Algorithm
• Dimensions of finitely generated FI-modules are eventually polynomial Corollary
• For an invertible matrix over a field, Cayley-Hamilton makes every matrix-power entry and trace sequence linearly recurrent Corollary
• Congruence need not preserve trace or determinant: the real 1×1 matrices [1] and [4] are congruent Counterexample
• Bloch coordinate maps Definition
• Dual Coxeter number for the normalized invariant form Definition
• Finite semidefinite primal-dual pair Definition
• Finite-basis projector energy model Definition
• Fixed-particle finite-temperature density-matrix free energy Definition
• Fixed-target large-margin nearest-neighbor objective Definition
• Matrix multiplicative-weights game and update Definition
• Parameterized variational quantum objective Definition
• Pauli measurement branch semantics Definition
• Quantum acceptance probability for a finite promise computation Definition
• Rank-r orthogonal projector domain Definition
• The fixed-particle admissible Hermitian density domain Definition
• Bloch coordinates of |0⟩ and H|0⟩ Example
• Matrix-unit rotation for a k–Mₙ(k) Morita pair Example
• Positive type on a discrete group: the identity mass, characters, and the regular GNS model Example
• Register endianness and a two-qubit partial trace Example
• The exterior square of the permutation FI-module has character polynomial C(X₁, 2)-X₂ Example
• The permutation FI-module has character polynomial X₁ Example
• A finite instrument with nonzero input has nonzero output space Lemma
• A Hermitian idempotent has trace equal to its rank Lemma
• A Hermitian two-by-two pivot inertia is determined by determinant and trace signs Lemma
• A strict occupied gap defines a unique rank-r spectral projector Lemma
• A uniform positive eigenvalue floor controls log-determinant error Lemma
• Additive characters are exactly one-dimensional complex representation characters Lemma
• Bounded independent matrix Bernstein bound from the trace exponential Lemma
• Complex projector tangents are skew-Hermitian commutator directions Lemma
• Finite matrix and module preliminaries: right inverses, rank invariance, finite length and nilpotent trace Lemma
• Finite real semidefinite programs satisfy strong duality under strict feasibility Lemma
• Finite-dimensional Golden–Thompson trace inequality Lemma
• Global Lipschitz ODE flows and the C1 density formula Lemma
• Graph charts and tangent directions for complex rank-r projectors Lemma
• Kullback–Leibler divergence between nondegenerate centered Gaussian laws Lemma
• Local separable trace-class determinant construction Lemma
• Normalized Hermitian inequality SDPs have equal primal and dual values Lemma
• Padded irreducible characters are eventually given by character polynomials Lemma
• Pauli measurement projectors and total probability Lemma

…and 39 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., Definition 8.47 (standard reference, not scraped)