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Sum of elements on the main diagonal
In linear algebra, the trace of a square matrix A, denoted tr(A), is defined as a sum of the elements on its main diagonal, a 11 + a 22 + ⋯ + a n n {\displaystyle
Trace_(linear_algebra)
is an outline of topics related to linear algebra, the branch of mathematics concerning linear equations and linear maps and their representations in vector
Outline_of_linear_algebra
Idempotent linear transformation from a vector space to itself
In linear algebra and functional analysis, a projection is a linear transformation P {\displaystyle P} from a vector space to itself (an endomorphism)
Projection_(linear_algebra)
Topics referred to by the same term
instruction fetch Trace table, a table used to test algorithms to ensure that no logical errors in calculation have occurred. Trace (linear algebra), the sum
Trace
Topics referred to by the same term
software update Stack pointer Stored procedure Synthetic programming Trace (linear algebra), or "Spur" (German), of a square matrix Sp(n) and Sp(2n,F), a symplectic
SP
Function over linear operators
In linear algebra and functional analysis, the partial trace is a generalization of the trace. Whereas the trace is a scalar-valued function on operators
Partial_trace
Group of 𝑛 × 𝑛 invertible matrices
resulting algebraic structure is a monoid, usually called the full linear monoid, but occasionally also full linear semigroup, general linear monoid etc
General_linear_group
Branch of mathematics that studies abstract algebraic structures
branch of mathematics that studies abstract algebraic structures by representing their elements as linear transformations of vector spaces. In essence
Representation_theory
Mathematical function
K-linear transformation of this vector space into itself. The trace, TrL/K(α), is defined as the trace (in the linear algebra sense) of this linear transformation
Field_trace
Norm on a vector space of matrices
Applied Numerical Linear Algebra, section 1.7, published by SIAM, 1997. Carl D. Meyer, Matrix Analysis and Applied Linear Algebra, published by SIAM
Matrix_norm
*-algebra of bounded operators on a Hilbert space
special cases states and traces are discussed in detail in (Takesaki 1979). A weight ω on a von Neumann algebra is a linear map from the set of positive
Von_Neumann_algebra
Topics referred to by the same term
Lockheed U-2 aircraft Technical Report, a type of ISO document Trace (linear algebra) Ton of refrigeration Total return Textus Receptus: Printed Greek
TR
Algebraic structure used in analysis
(u)\operatorname {ad} (v)),} where tr denotes the trace of a linear operator. Namely: a Lie algebra g {\displaystyle {\mathfrak {g}}} is semisimple if
Lie_algebra
Graphical means of performing computations in linear algebra
In mathematics, trace diagrams are a graphical means of performing computations in linear and multilinear algebra. They can be represented as (slightly
Trace_diagram
Concept in mathematics
In mathematics, the special linear Lie algebra of order n {\displaystyle n} over a field F {\displaystyle F} , denoted s l n F {\displaystyle {\mathfrak
Special_linear_Lie_algebra
Compact operator for which a finite trace can be defined
used to compute the trace. This trace of trace-class operators generalizes the trace of matrices studied in linear algebra. All trace-class operators are
Trace_class
Number of vectors in any basis of the vector space
a notion of dimension when one has a trace but no natural sense of basis. For example, one may have an algebra A {\displaystyle A} with maps η : K →
Dimension_(vector_space)
Equivalence under a change of basis (linear algebra)
In linear algebra, two n-by-n matrices A and B are called similar if there exists an invertible n-by-n matrix P such that B = P − 1 A P . {\displaystyle
Matrix_similarity
Group of matrices with determinant 1
field). These elements are "special" in that they form an algebraic subvariety of the general linear group – they satisfy a polynomial equation (since the
Special_linear_group
In mathematics, the trace field of a linear group is the field generated by the traces of its elements. It is mostly studied for Kleinian and Fuchsian
Trace field of a representation
Trace_field_of_a_representation
Algebraic structure in mathematics
In mathematics, a quadratic algebra is an algebra over a ring for which the algebra extends the ring by a new element that satisfies a monic, quadratic
Quadratic_algebra
Noncommutative geometric structure
singular trace is a trace on a space of linear operators of a separable Hilbert space that vanishes on operators of finite rank. Singular traces are a feature
Singular_trace
In mathematics, invariant of square matrices
that its Lie algebra is the special linear Lie algebra s l n {\displaystyle {\mathfrak {sl}}_{n}} consisting of those matrices having trace zero. Writing
Determinant
Array of numbers
a 2 × 3 matrix, or a matrix of dimension 2 × 3. In linear algebra, matrices are used as linear maps. In geometry, matrices are used for geometric transformations
Matrix_(mathematics)
Differential algebra
In abstract algebra, the Weyl algebras are abstracted from the ring of differential operators with polynomial coefficients. They are named after Hermann
Weyl_algebra
Similar to the basis of a vector space, but not necessarily linearly independent
In linear algebra, a frame of an inner product space is a generalization of a basis of a vector space to sets that may be linearly dependent. In the terminology
Frame_(linear_algebra)
Algebraic structure with "nice" duality properties
bilinear form is called the Frobenius form of the algebra. Equivalently, one may equip A with a linear functional λ : A → k such that the kernel of λ contains
Frobenius_algebra
Concepts from linear algebra
In linear algebra, an eigenvector (/ˈaɪɡən-/ EYE-gən-) or characteristic vector is a (nonzero) vector that has its direction unchanged (or reversed) by
Eigenvalues_and_eigenvectors
Group representation
mathematics and theoretical physics, a representation of a Lie group is a linear action of a Lie group on a vector space. Equivalently, a representation
Representation_of_a_Lie_group
Properties of mathematical relationships
transformation). Linear algebra is the branch of mathematics concerned with systems of linear equations. In Boolean algebra, a linear function is a function
Linearity
Mathematical operation in linear algebra
In mathematics, specifically in linear algebra, matrix multiplication is a binary operation that produces a matrix from two matrices. For matrix multiplication
Matrix_multiplication
Identifies the commutant of a specific von Neumann algebra
theorem for traces explicitly identifies the commutant of a specific von Neumann algebra acting on a Hilbert space in the presence of a trace. The first
Commutation theorem for traces
Commutation_theorem_for_traces
Polynomial whose roots are the eigenvalues of a matrix
In linear algebra, the characteristic polynomial of a square matrix is a polynomial which is invariant under matrix similarity and has the eigenvalues
Characteristic_polynomial
Linear map from a vector space to its field of scalars
(Terse) Introduction to Linear Algebra, American Mathematical Society, ISBN 978-0-8218-4419-9 Lax, Peter (1996), Linear algebra, Wiley-Interscience,
Linear_form
C*-algebra of complex square matrices with the positive elements being the positive-definite matrices. The trace function defined on this C*-algebra is
Positive_linear_functional
Mathematical term
way of representing the elements of the group as linear transformations of the group's Lie algebra, considered as a vector space. For example, if G is
Adjoint_representation
248-dimensional exceptional simple Lie group
several closely related exceptional simple Lie groups, linear algebraic groups or Lie algebras of dimension 248; the same notation is used for the corresponding
E8_(mathematics)
a central role. Abstract algebra is largely a product of the 19th and 20th centuries. The origins of algebra can be traced to the ancient Babylonians
History_of_algebra
Associative algebra introduced by Richard Brauer
does for the representation theory of the general linear group in Schur–Weyl duality. The Brauer algebra B n ( δ ) {\displaystyle {\mathfrak {B}}_{n}(\delta
Brauer_algebra
Fundamental theorem in probability theory and statistics
coefficients), M ↦ tr(AM) where A is the matrix of the coefficients; see Trace (linear algebra)#Inner product. A random orthogonal matrix is said to be distributed
Central_limit_theorem
characterizing properties of finite von Neumann algebras is the existence of a center-valued trace. A von Neumann algebra M {\displaystyle {\mathcal {M}}} is finite
Finite_von_Neumann_algebra
Nilpotent subalgebra of a Lie algebra
Lie algebra of a maximal torus of the compact group. If g {\displaystyle {\mathfrak {g}}} is a linear Lie algebra (a Lie subalgebra of the Lie algebra of
Cartan_subalgebra
Operation in mathematics
In multilinear algebra, a tensor contraction is an operation on a tensor that arises from the canonical pairing of a vector space and its dual. This example
Tensor_contraction
Mathematical object studied in the field of algebraic geometry
Algebraic varieties are the central objects of study in algebraic geometry, a sub-field of mathematics. Classically, an algebraic variety is defined as
Algebraic_variety
Non-associative algebras with positive-definite quadratic form
algebra H3(O) is called the Albert algebra after A.A. Albert. To check that Hn(A) satisfies the axioms for a Euclidean Jordan algebra, the real trace
Hurwitz's theorem (composition algebras)
Hurwitz's_theorem_(composition_algebras)
Type of vector space in math
same thing as its dimension as a linear space (the cardinality of a Hamel basis). Koashi, Masato, "Appendix: Linear algebra" (PDF) Hewitt & Stromberg (1965
Hilbert_space
Branch of mathematics
whose origins can be traced to investigations in combinatorial topology (a precursor to algebraic topology) and abstract algebra (theory of modules and
Homological_algebra
In linear algebra, similarity invariance is a property exhibited by a function whose value is unchanged under similarities of its domain. That is, f {\displaystyle
Similarity_invariance
Group of real 2×2 matrices with unit determinant
its Lie algebra sl(2, R) by conjugation (remember that the Lie algebra elements are also 2 × 2 matrices), yielding a faithful 3-dimensional linear representation
SL2(R)
real-linear finite-dimensional representation of a real Lie algebra extends to a complex-linear representation of its complexification. The real-linear representation
Representation theory of semisimple Lie algebras
Representation_theory_of_semisimple_Lie_algebras
complex semisimple Lie groups and semisimple Lie algebras. An element X of a semisimple Lie algebra g is called nilpotent if its adjoint endomorphism
Nilpotent_orbit
Algebraic object with geometric applications
In mathematics, a tensor is an algebraic object that describes a multilinear relationship between sets of algebraic objects associated with a vector space
Tensor
Shape that can be represented as a linear matrix inequality
convex geometry, a spectrahedron is a shape that can be represented as a linear matrix inequality. Alternatively, the set of n × n positive semidefinite
Spectrahedron
Open convex self-dual cones
C* algebra. The complexification of a simple Euclidean Jordan algebra is a simple complex Jordan algebra which is also separable, i.e. its trace form
Symmetric_cone
Group of unitary complex matrices with determinant of 1
Lie algebra s u ( n ) {\displaystyle {\mathfrak {su}}(n)} of SU(n) consists of n × n skew-Hermitian matrices with trace zero. This (real) Lie algebra has
Special_unitary_group
Type of mathematical system
Hirsch, Morris W. (1974). Differential equations, dynamical systems, and linear algebra. Pure and applied mathematics. Stephen Smale. New York: Academic Press
Linear_dynamical_system
Theorem
representation of Φ. M.-D. Choi, Completely Positive Linear Maps on Complex Matrices, Linear Algebra and its Applications, 10, 285–290 (1975). V. P. Belavkin
Stinespring_dilation_theorem
1957 book by Emil Artin
students studying mathematics. From the Preface: Linear algebra, topology, differential and algebraic geometry are the indispensable tools of the mathematician
Geometric_Algebra_(book)
Finite extension of the rationals
In mathematics, an algebraic number field (or simply number field) is an extension field K {\displaystyle K} of the field of rational numbers Q {\displaystyle
Algebraic_number_field
Mathematical group
precisely the linear changes of variables that preserve the standard symplectic form, or equivalently the Poisson bracket. The Lie algebra s p ( 2 n , R
Symplectic_group
Equivalence of von Neumann Algebras Chapter 8. The Trace Chapter 9. Algebra and Commutant Chapter 10. Special Representation of C*-Algebras Chapter 11. Tensor
Fundamentals of the Theory of Operator Algebras
Fundamentals_of_the_Theory_of_Operator_Algebras
universal enveloping algebra of g 1 {\displaystyle {\mathfrak {g}}_{1}} . Via π {\displaystyle \pi } , it acts on V as a linear endomorphism (namely,
Whitehead's lemma (Lie algebra)
Whitehead's_lemma_(Lie_algebra)
Vector space with generalized dot product
authors, especially in physics and matrix algebra, prefer to define inner products and sesquilinear forms with linearity in the second argument rather than the
Inner_product_space
Symmetric bilinear form in mathematics
coefficients of the Lie algebra. The index k functions as column index and the index n as row index in the matrix ad(ei)ad(ej). Taking the trace amounts to putting
Killing_form
Type of continuous linear operator
compact closure. Compactness is therefore automatic in finite-dimensional linear algebra. Conversely, the identity operator on an infinite-dimensional Banach
Compact_operator
Vector satisfying some of the criteria of an eigenvector
In linear algebra, a generalized eigenvector of an n × n {\displaystyle n\times n} matrix A {\displaystyle A} is a vector which satisfies certain criteria
Generalized_eigenvector
Mathematical ring whose elements are matrices
matrices. A matrix ring over a field is a Frobenius algebra, with Frobenius form given by the trace of the product: σ(A, B) = tr(AB). The matrix ring Mn(R)
Matrix_ring
Families of matrices in mathematics, physics, and quantum information
Pauli matrices refers to families of matrices which generalize the (linear algebraic) properties of the Pauli matrices. Here, a few classes of such matrices
Generalizations of Pauli matrices
Generalizations_of_Pauli_matrices
Theorem relating to matrix equivalence
Charles R. (2007), "Unitarily achievable zero patterns and traces of words in A and A*", Linear Algebra and Its Applications, 421 (1): 63–68, doi:10.1016/j.laa
Specht's_theorem
Theorem in linear algebra
In mathematics, particularly in linear algebra, the Schur product theorem states that the Hadamard product of two positive definite matrices is also a
Schur_product_theorem
Set of a matrix's eigenvalues
(3rd ed.), New York: Wiley, ISBN 0-471-50728-8 Nering, Evar D. (1970), Linear Algebra and Matrix Theory (2nd ed.), New York: Wiley, LCCN 76091646 v t e
Spectrum_of_a_matrix
{\mathfrak {g}}} in the upper triangular form over the algebraic closure of the ground field (the trace can be computed after extending the ground field).
Cartan's_criterion
Theorem representing a solvable Lie algebra
algebra (see #Consequences). Also, to each flag in a finite-dimensional vector space V, there correspond a Borel subalgebra (that consist of linear transformations
Lie's_theorem
Branch of mathematics that studies algebraic structures
algebra in Wiktionary, the free dictionary. In mathematics, more specifically algebra, abstract algebra or modern algebra is the study of algebraic structures
List of abstract algebra topics
List_of_abstract_algebra_topics
Mathematical concept in algebra
In linear algebra, a nilpotent matrix is a square matrix N such that N k = 0 {\displaystyle N^{k}=0\,} for some positive integer k {\displaystyle k} .
Nilpotent_matrix
Theorem in abstract algebra
abstract algebra, the Frobenius theorem, proved by Ferdinand Georg Frobenius in 1877, characterizes the finite-dimensional associative division algebras over
Frobenius theorem (real division algebras)
Frobenius_theorem_(real_division_algebras)
Matrices important in quantum mechanics and the study of spin
{a}}|)=0~.} A standard result in linear algebra (a linear map that satisfies a polynomial equation written in distinct linear factors is diagonalizable) means
Pauli_matrices
Irish mathematician
(electronic). 1999 "A characterization of trace zero nonnegative 5 × 5 matrices". (with E. Meehan) Linear Algebra Appl., 302–3. 1996 "The real and the
Thomas_J._Laffey
Concept in linear algebra
In linear algebra, a Householder transformation (also known as a Householder reflection or elementary reflector) is a linear transformation that describes
Householder_transformation
generators of the special linear Lie algebra sl2. This notion plays an important role in the theory of semisimple Lie algebras, especially in regard to
Sl2-triple
C*-algebra Positive element Positive linear functional operator algebra nest algebra reflexive operator algebra Calkin algebra Gelfand representation Gelfand–Naimark
List of functional analysis topics
List_of_functional_analysis_topics
Representation theory
is part of the Peter–Weyl theorem); so the notion of trace is the usual one from linear algebra. Knowledge of the character χ {\displaystyle \chi } of
Weyl_character_formula
Basis for the SU(3) Lie algebra
eight linearly independent 3×3 traceless Hermitian matrices used in the study of the strong interaction in particle physics. They span the Lie algebra of
Gell-Mann_matrices
Algebraic structure used in theoretical physics
Lie algebra as all the signs disappear, and the superbracket becomes a normal Lie bracket, while g 1 {\displaystyle {\mathfrak {g}}_{1}} is a linear representation
Lie_superalgebra
Extension of Lidskii's theorem
spaces. The theorem should not be confused with the Grothendieck trace formula from algebraic geometry. Given a Banach space ( B , ‖ ⋅ ‖ ) {\displaystyle (B
Grothendieck_trace_theorem
(channels) from A to Cn×n, where A is a C*-algebra and Cn×n denotes the n×n complex entries, and positive linear functionals (states) on the tensor product
Channel-state_duality
Generalization of matrix trace
multiplication by the trace of the endomorphism f in the usual sense of linear algebra. More generally, in the category of modules over a ring R, the dualizable
Categorical_trace
Linear operator in functional analysis
setting. As such, these operators may be described via linear algebra techniques. From linear algebra, we know that a rectangular matrix, with complex entries
Finite-rank_operator
there will be non-linear conditions for a tensor to satisfy, to be pure. For more see Segre embedding. Tensor algebra In the tensor algebra T(V) of a vector
Glossary_of_tensor_theory
Square matrix with ones on the main diagonal and zeros elsewhere
In linear algebra, the identity matrix of size n {\displaystyle n} is the n × n {\displaystyle n\times n} square matrix with ones on the main diagonal
Identity_matrix
Graphical notation for multilinear algebra calculations
with the presence of matrix groups to trace diagrams in linear algebra. In the language of multilinear algebra, each shape represents a multilinear function
Penrose_graphical_notation
Mapping theorem in topology
(H_{k}(f,\mathbb {Q} )),} the alternating (finite) sum of the matrix traces of the linear maps induced by f {\displaystyle f} on H k ( X , Q ) {\displaystyle
Lefschetz_fixed-point_theorem
tan, tg.) Thm – theorem. Tor – Tor functor. Tr – field trace. tr – trace of a matrix or linear transformation. (Also written as Sp.) undef – a function
List of mathematical abbreviations
List_of_mathematical_abbreviations
Matrix decomposition
In linear algebra, the singular value decomposition (SVD) is a factorization of a real or complex matrix into a rotation, followed by a scaling, followed
Singular_value_decomposition
Low-rank isomorphisms in mathematics
isomorphisms of the associated Lie algebras. These exceptional isomorphisms can be constructed by exhibiting small linear representations that preserve a
Exceptional isomorphisms of classical groups
Exceptional_isomorphisms_of_classical_groups
Reals with an extra square root of +1 adjoined
In algebra, a split-complex number (or hyperbolic number, also perplex number, double number) is based on a hyperbolic unit j satisfying j 2 = 1 {\displaystyle
Split-complex_number
This is a glossary of algebraic geometry. See also glossary of commutative algebra, glossary of classical algebraic geometry, and glossary of ring theory
Glossary of algebraic geometry
Glossary_of_algebraic_geometry
Monster and modular connection
structure of a super vertex algebra over Fp, and they broke up the problem into an easy step equating graded Brauer super-trace with the McKay-Thompson series
Monstrous_moonshine
For a square matrix, the transpose of the cofactor matrix
In linear algebra, the adjugate or classical adjoint adj(A) of a square matrix A is the transpose of its cofactor matrix. It is occasionally known as adjunct
Adjugate_matrix
Lie group of Lorentz transformations
the inhomogeneous Lorentz group. Lorentz transformations are examples of linear transformations; general isometries of Minkowski spacetime are affine transformations
Lorentz_group
travel, tourism, insurance
TRACE LINEAR-ALGEBRA
TRACE LINEAR-ALGEBRA
Female
English
Variant spelling of English Linsey, LINSAY means "Lincoln's wetlands."
Female
English
Feminine variant spelling of English unisex Tracy, TRACEE means "place of Thracius."
Male
English
Variant spelling of English unisex Tracy, TRACEY means "place of Thracius."
Female
Scottish
Variant spelling of Scottish Lilias, LILEAS means "lily."
Surname or Lastname
English
English : perhaps a variant of Treece.
Male
English
English surname transferred to unisex forename use, from a Norman baronial name TRACY means "place of Thracius."
Surname or Lastname
English
English : nickname from Middle English, Old French grace ‘charm’, ‘pleasantness’ (Latin gratia).English : from the female personal name Grace, which was popular in the Middle Ages. This seems in the first instance to have been from a Germanic element grīs ‘gray’ (see Grice 1), but was soon associated by folk etymology with the Latin word meaning ‘charm’.
Surname or Lastname
English
English : variant of Lingard.French : occupational name for a maker of or dealer in linen goods, from Old French linge ‘linen (goods)’ (see Linge 1).
Girl/Female
Latin American English Irish
Grace.
Female
English
Feminine variant spelling of English unisex Tracy, TRACIE means "place of Thracius."
Girl/Female
American, Arabic, Australian, British, Chinese, Christian, Danish, English, French, German, Gujarati, Indian, Irish, Jamaican, Latin, Muslim, Portuguese, Swedish
Mercy; God's Favor; Grace; Grace of God; Kindness; Thanks; Love; Favour; Blessing; Charm; Good will
Female
English
Feminine variant spelling of English unisex Tracy, TRACI means "place of Thracius."
Male
Greek
(ΑἰνÎας) Variant spelling of Greek AineÃas, AINEAS means "praiseworthy."
Boy/Male
Hindu
Lingam
Male
English
Irish Anglicized form of Gaelic Fionnbarr, FINBAR means "fair-headed."
Boy/Male
Anglo Saxon American English French
Brave.
Boy/Male
American, Anglo, Australian, British, Chinese, English, French
Fighter; Brave
Surname or Lastname
English
English : probably from Middle English, Old French brace ‘arm’, also denoting a piece of armor covering the arm. In most cases it is probably a metonymic occupational name for a maker or seller of armor, specifically armor designed to protect the upper arms, but it could also have been a nickname for someone with strong arms (compare Armstrong) or a deformed or otherwise noticeable arm.
Male
Yiddish
 Variant spelling of Yiddish Lieber, LIBER means "beloved." Compare with another form of Liber.
Male
English
Short form of English unisex Tracy, TRACE means "place of Thracius."
TRACE LINEAR-ALGEBRA
TRACE LINEAR-ALGEBRA
TRACE LINEAR-ALGEBRA
TRACE LINEAR-ALGEBRA
TRACE LINEAR-ALGEBRA
TRACE LINEAR-ALGEBRA
TRACE LINEAR-ALGEBRA
v. t.
To follow the tracks or traces of; to pursue by following the marks of the feet; to trace; to trail; as, to track a deer in the snow.
n.
A tract or area, as of land.
n.
One who, or that which, traces.
v. t.
To mark out; to draw or delineate with marks; especially, to copy, as a drawing or engraving, by following the lines and marking them on a sheet superimposed, through which they appear; as, to trace a figure or an outline; a traced drawing.
v. t.
To trace out; to track; also, to draw out; to protact.
v. t.
Hence, to follow the trace or track of.
a.
Of or pertaining to a line; consisting of lines; in a straight direction; lineal.
a.
Descending in a direct line from an ancestor; hereditary; derived from ancestors; -- opposed to collateral; as, a lineal descent or a lineal descendant.
n.
One who lines, as, a liner of shoes.
a.
Composed of lines; delineated; as, lineal designs.
a.
In the direction of a line; of or pertaining to a line; measured on, or ascertained by, a line; linear; as, lineal magnitude.
v. t.
To add grace notes, cadenzas, etc., to.
a.
Like a line; narrow; of the same breadth throughout, except at the extremities; as, a linear leaf.
a.
Linear.
v.
Track; trace.
v.
The trade winds.
imp. & p. p.
of Trace
v. t.
To supply with heavenly grace.
adv.
In a linear manner; with lines.
v. t.
A mark left by anything passing; a track; a path; a course; a footprint; a vestige; as, the trace of a carriage or sled; the trace of a deer; a sinuous trace.
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