Logo
Unionpedia
Communication
Get it on Google Play
New! Download Unionpedia on your Android™ device!
Download
Faster access than browser!
 

Trace (linear algebra)

Index Trace (linear algebra)

In linear algebra, the trace of an n-by-n square matrix A is defined to be the sum of the elements on the main diagonal (the diagonal from the upper left to the lower right) of A, i.e., where aii denotes the entry on the ith row and ith column of A. The trace of a matrix is the sum of the (complex) eigenvalues, and it is invariant with respect to a change of basis. [1]

91 relations: Associative algebra, Basis (linear algebra), Bialgebra, Bilinear form, Bounded operator, Cauchy–Schwarz inequality, Change of basis, Character (mathematics), Characteristic polynomial, Coalgebra, Commutator, Compact operator, Complex number, Cyclic permutation, Derivative, Determinant, Diagonal matrix, Dimension (vector space), Direct sum of modules, Divergence, Divergence theorem, Dot product, Dual space, Eigenvalues and eigenvectors, Exact sequence, Field trace, Flow network, Golden–Thompson inequality, Group representation, Hadamard product (matrices), Hermitian matrix, Hilbert space, Hilbert–Schmidt operator, Idempotent matrix, Identity matrix, Inner product space, Invariants of tensors, Jacobi's formula, Jordan normal form, Killing form, Kronecker delta, Kronecker product, Lie algebra, Lie group, Linear algebra, Linear form, Linear map, Main diagonal, Matrix (mathematics), Matrix calculus, ..., Matrix exponential, Matrix multiplication, Matrix norm, Matrix similarity, Möbius transformation, Natural transformation, Nilpotent matrix, Norm (mathematics), Orthonormal basis, Partial trace, Positive-definite matrix, Projection matrix, Quadratic form (statistics), Rank (linear algebra), Real number, Representation theory, Scalar (mathematics), Semisimple Lie algebra, Similarity invariance, Simple Lie group, Skew-symmetric matrix, Solvable Lie algebra, Specht's theorem, Special linear group, Special linear Lie algebra, Square matrix, Statistics, Superalgebra, Supertrace, Symmetric matrix, Tensor contraction, Tensor product, Trace class, Trace inequalities, Traced monoidal category, Transpose, Triangular matrix, Vector space, Vectorization (mathematics), Volume, 0. Expand index (41 more) »

Associative algebra

In mathematics, an associative algebra is an algebraic structure with compatible operations of addition, multiplication (assumed to be associative), and a scalar multiplication by elements in some field.

New!!: Trace (linear algebra) and Associative algebra · See more »

Basis (linear algebra)

In mathematics, a set of elements (vectors) in a vector space V is called a basis, or a set of, if the vectors are linearly independent and every vector in the vector space is a linear combination of this set.

New!!: Trace (linear algebra) and Basis (linear algebra) · See more »

Bialgebra

In mathematics, a bialgebra over a field K is a vector space over K which is both a unital associative algebra and a coalgebra.

New!!: Trace (linear algebra) and Bialgebra · See more »

Bilinear form

In mathematics, more specifically in abstract algebra and linear algebra, a bilinear form on a vector space V is a bilinear map, where K is the field of scalars.

New!!: Trace (linear algebra) and Bilinear form · See more »

Bounded operator

In functional analysis, a bounded linear operator is a linear transformation L between normed vector spaces X and Y for which the ratio of the norm of L(v) to that of v is bounded above by the same number, over all non-zero vectors v in X. In other words, there exists some M\ge 0 such that for all v in X The smallest such M is called the operator norm \|L\|_ \, of L. A bounded linear operator is generally not a bounded function; the latter would require that the norm of L(v) be bounded for all v, which is not possible unless L(v).

New!!: Trace (linear algebra) and Bounded operator · See more »

Cauchy–Schwarz inequality

In mathematics, the Cauchy–Schwarz inequality, also known as the Cauchy–Bunyakovsky–Schwarz inequality, is a useful inequality encountered in many different settings, such as linear algebra, analysis, probability theory, vector algebra and other areas.

New!!: Trace (linear algebra) and Cauchy–Schwarz inequality · See more »

Change of basis

In linear algebra, a basis for a vector space of dimension n is a set of n vectors, called basis vectors, with the property that every vector in the space can be expressed as a unique linear combination of the basis vectors.

New!!: Trace (linear algebra) and Change of basis · See more »

Character (mathematics)

In mathematics, a character is (most commonly) a special kind of function from a group to a field (such as the complex numbers).

New!!: Trace (linear algebra) and Character (mathematics) · See more »

Characteristic polynomial

In linear algebra, the characteristic polynomial of a square matrix is a polynomial which is invariant under matrix similarity and has the eigenvalues as roots.

New!!: Trace (linear algebra) and Characteristic polynomial · See more »

Coalgebra

In mathematics, coalgebras or cogebras are structures that are dual (in the category-theoretic sense of reversing arrows) to unital associative algebras.

New!!: Trace (linear algebra) and Coalgebra · See more »

Commutator

In mathematics, the commutator gives an indication of the extent to which a certain binary operation fails to be commutative.

New!!: Trace (linear algebra) and Commutator · See more »

Compact operator

In functional analysis, a branch of mathematics, a compact operator is a linear operator L from a Banach space X to another Banach space Y, such that the image under L of any bounded subset of X is a relatively compact subset (has compact closure) of Y. Such an operator is necessarily a bounded operator, and so continuous.

New!!: Trace (linear algebra) and Compact operator · See more »

Complex number

A complex number is a number that can be expressed in the form, where and are real numbers, and is a solution of the equation.

New!!: Trace (linear algebra) and Complex number · See more »

Cyclic permutation

In mathematics, and in particular in group theory, a cyclic permutation (or cycle) is a permutation of the elements of some set X which maps the elements of some subset S of X to each other in a cyclic fashion, while fixing (that is, mapping to themselves) all other elements of X. If S has k elements, the cycle is called a k-cycle.

New!!: Trace (linear algebra) and Cyclic permutation · See more »

Derivative

The derivative of a function of a real variable measures the sensitivity to change of the function value (output value) with respect to a change in its argument (input value).

New!!: Trace (linear algebra) and Derivative · See more »

Determinant

In linear algebra, the determinant is a value that can be computed from the elements of a square matrix.

New!!: Trace (linear algebra) and Determinant · See more »

Diagonal matrix

In linear algebra, a diagonal matrix is a matrix in which the entries outside the main diagonal are all zero.

New!!: Trace (linear algebra) and Diagonal matrix · See more »

Dimension (vector space)

In mathematics, the dimension of a vector space V is the cardinality (i.e. the number of vectors) of a basis of V over its base field.

New!!: Trace (linear algebra) and Dimension (vector space) · See more »

Direct sum of modules

In abstract algebra, the direct sum is a construction which combines several modules into a new, larger module.

New!!: Trace (linear algebra) and Direct sum of modules · See more »

Divergence

In vector calculus, divergence is a vector operator that produces a scalar field, giving the quantity of a vector field's source at each point.

New!!: Trace (linear algebra) and Divergence · See more »

Divergence theorem

In vector calculus, the divergence theorem, also known as Gauss's theorem or Ostrogradsky's theorem, reprinted in is a result that relates the flow (that is, flux) of a vector field through a surface to the behavior of the vector field inside the surface.

New!!: Trace (linear algebra) and Divergence theorem · See more »

Dot product

In mathematics, the dot product or scalar productThe term scalar product is often also used more generally to mean a symmetric bilinear form, for example for a pseudo-Euclidean space.

New!!: Trace (linear algebra) and Dot product · See more »

Dual space

In mathematics, any vector space V has a corresponding dual vector space (or just dual space for short) consisting of all linear functionals on V, together with the vector space structure of pointwise addition and scalar multiplication by constants.

New!!: Trace (linear algebra) and Dual space · See more »

Eigenvalues and eigenvectors

In linear algebra, an eigenvector or characteristic vector of a linear transformation is a non-zero vector that changes by only a scalar factor when that linear transformation is applied to it.

New!!: Trace (linear algebra) and Eigenvalues and eigenvectors · See more »

Exact sequence

An exact sequence is a concept in mathematics, especially in group theory, ring and module theory, homological algebra, as well as in differential geometry.

New!!: Trace (linear algebra) and Exact sequence · See more »

Field trace

In mathematics, the field trace is a particular function defined with respect to a finite field extension L/K, which is a ''K''-linear map from L onto K.

New!!: Trace (linear algebra) and Field trace · See more »

Flow network

In graph theory, a flow network (also known as a transportation network) is a directed graph where each edge has a capacity and each edge receives a flow.

New!!: Trace (linear algebra) and Flow network · See more »

Golden–Thompson inequality

In physics and mathematics, the Golden–Thompson inequality is a trace inequality between exponentials of matrices proved independently by and.

New!!: Trace (linear algebra) and Golden–Thompson inequality · See more »

Group representation

In the mathematical field of representation theory, group representations describe abstract groups in terms of linear transformations of vector spaces; in particular, they can be used to represent group elements as matrices so that the group operation can be represented by matrix multiplication.

New!!: Trace (linear algebra) and Group representation · See more »

Hadamard product (matrices)

In mathematics, the Hadamard product (also known as the Schur product or the entrywise product) is a binary operation that takes two matrices of the same dimensions, and produces another matrix where each element i,j is the product of elements i,j of the original two matrices.

New!!: Trace (linear algebra) and Hadamard product (matrices) · See more »

Hermitian matrix

In mathematics, a Hermitian matrix (or self-adjoint matrix) is a complex square matrix that is equal to its own conjugate transpose—that is, the element in the -th row and -th column is equal to the complex conjugate of the element in the -th row and -th column, for all indices and: Hermitian matrices can be understood as the complex extension of real symmetric matrices.

New!!: Trace (linear algebra) and Hermitian matrix · See more »

Hilbert space

The mathematical concept of a Hilbert space, named after David Hilbert, generalizes the notion of Euclidean space.

New!!: Trace (linear algebra) and Hilbert space · See more »

Hilbert–Schmidt operator

In mathematics, a Hilbert–Schmidt operator, named for David Hilbert and Erhard Schmidt, is a bounded operator A on a Hilbert space H with finite Hilbert–Schmidt norm where \|\ \| is the norm of H, \ an orthonormal basis of H, and Tr is the trace of a nonnegative self-adjoint operator.

New!!: Trace (linear algebra) and Hilbert–Schmidt operator · See more »

Idempotent matrix

In linear algebra, an idempotent matrix is a matrix which, when multiplied by itself, yields itself.

New!!: Trace (linear algebra) and Idempotent matrix · See more »

Identity matrix

In linear algebra, the identity matrix, or sometimes ambiguously called a unit matrix, of size n is the n × n square matrix with ones on the main diagonal and zeros elsewhere.

New!!: Trace (linear algebra) and Identity matrix · See more »

Inner product space

In linear algebra, an inner product space is a vector space with an additional structure called an inner product.

New!!: Trace (linear algebra) and Inner product space · See more »

Invariants of tensors

In mathematics, in the fields of multilinear algebra and representation theory, invariants of tensors are coefficients of the characteristic polynomial of the tensor A: where \mathbf is the identity tensor and \lambda\in\mathbb is the polynomial's indeterminate (it is important to bear in mind that a polynomial's indeterminate \lambda may also be a non-scalar as long as power, scaling and adding make sense for it, e.g., p(\mathbf) is legitimate, and in fact, quite useful).

New!!: Trace (linear algebra) and Invariants of tensors · See more »

Jacobi's formula

In matrix calculus, Jacobi's formula expresses the derivative of the determinant of a matrix A in terms of the adjugate of A and the derivative of A. If is a differentiable map from the real numbers to matrices, where is the trace of the matrix.

New!!: Trace (linear algebra) and Jacobi's formula · See more »

Jordan normal form

In linear algebra, a Jordan normal form (often called Jordan canonical form) of a linear operator on a finite-dimensional vector space is an upper triangular matrix of a particular form called a Jordan matrix, representing the operator with respect to some basis.

New!!: Trace (linear algebra) and Jordan normal form · See more »

Killing form

In mathematics, the Killing form, named after Wilhelm Killing, is a symmetric bilinear form that plays a basic role in the theories of Lie groups and Lie algebras.

New!!: Trace (linear algebra) and Killing form · See more »

Kronecker delta

In mathematics, the Kronecker delta (named after Leopold Kronecker) is a function of two variables, usually just non-negative integers.

New!!: Trace (linear algebra) and Kronecker delta · See more »

Kronecker product

In mathematics, the Kronecker product, denoted by ⊗, is an operation on two matrices of arbitrary size resulting in a block matrix.

New!!: Trace (linear algebra) and Kronecker product · See more »

Lie algebra

In mathematics, a Lie algebra (pronounced "Lee") is a vector space \mathfrak g together with a non-associative, alternating bilinear map \mathfrak g \times \mathfrak g \rightarrow \mathfrak g; (x, y) \mapsto, called the Lie bracket, satisfying the Jacobi identity.

New!!: Trace (linear algebra) and Lie algebra · See more »

Lie group

In mathematics, a Lie group (pronounced "Lee") is a group that is also a differentiable manifold, with the property that the group operations are compatible with the smooth structure.

New!!: Trace (linear algebra) and Lie group · See more »

Linear algebra

Linear algebra is the branch of mathematics concerning linear equations such as linear functions such as and their representations through matrices and vector spaces.

New!!: Trace (linear algebra) and Linear algebra · See more »

Linear form

In linear algebra, a linear functional or linear form (also called a one-form or covector) is a linear map from a vector space to its field of scalars.

New!!: Trace (linear algebra) and Linear form · See more »

Linear map

In mathematics, a linear map (also called a linear mapping, linear transformation or, in some contexts, linear function) is a mapping between two modules (including vector spaces) that preserves (in the sense defined below) the operations of addition and scalar multiplication.

New!!: Trace (linear algebra) and Linear map · See more »

Main diagonal

In linear algebra, the main diagonal (sometimes principal diagonal, primary diagonal, leading diagonal, or major diagonal) of a matrix A is the collection of entries A_ where i.

New!!: Trace (linear algebra) and Main diagonal · See more »

Matrix (mathematics)

In mathematics, a matrix (plural: matrices) is a rectangular array of numbers, symbols, or expressions, arranged in rows and columns.

New!!: Trace (linear algebra) and Matrix (mathematics) · See more »

Matrix calculus

In mathematics, matrix calculus is a specialized notation for doing multivariable calculus, especially over spaces of matrices.

New!!: Trace (linear algebra) and Matrix calculus · See more »

Matrix exponential

In mathematics, the matrix exponential is a matrix function on square matrices analogous to the ordinary exponential function.

New!!: Trace (linear algebra) and Matrix exponential · See more »

Matrix multiplication

In mathematics, matrix multiplication or matrix product is a binary operation that produces a matrix from two matrices with entries in a field, or, more generally, in a ring or even a semiring.

New!!: Trace (linear algebra) and Matrix multiplication · See more »

Matrix norm

In mathematics, a matrix norm is a vector norm in a vector space whose elements (vectors) are matrices (of given dimensions).

New!!: Trace (linear algebra) and Matrix norm · See more »

Matrix similarity

In linear algebra, two n-by-n matrices and are called similar if for some invertible n-by-n matrix.

New!!: Trace (linear algebra) and Matrix similarity · See more »

Möbius transformation

In geometry and complex analysis, a Möbius transformation of the complex plane is a rational function of the form of one complex variable z; here the coefficients a, b, c, d are complex numbers satisfying ad − bc ≠ 0.

New!!: Trace (linear algebra) and Möbius transformation · See more »

Natural transformation

In category theory, a branch of mathematics, a natural transformation provides a way of transforming one functor into another while respecting the internal structure (i.e., the composition of morphisms) of the categories involved.

New!!: Trace (linear algebra) and Natural transformation · See more »

Nilpotent matrix

In linear algebra, a nilpotent matrix is a square matrix N such that for some positive integer k. The smallest such k is sometimes called the index of N. More generally, a nilpotent transformation is a linear transformation L of a vector space such that Lk.

New!!: Trace (linear algebra) and Nilpotent matrix · See more »

Norm (mathematics)

In linear algebra, functional analysis, and related areas of mathematics, a norm is a function that assigns a strictly positive length or size to each vector in a vector space—save for the zero vector, which is assigned a length of zero.

New!!: Trace (linear algebra) and Norm (mathematics) · See more »

Orthonormal basis

In mathematics, particularly linear algebra, an orthonormal basis for an inner product space V with finite dimension is a basis for V whose vectors are orthonormal, that is, they are all unit vectors and orthogonal to each other.

New!!: Trace (linear algebra) and Orthonormal basis · See more »

Partial trace

In linear algebra and functional analysis, the partial trace is a generalization of the trace.

New!!: Trace (linear algebra) and Partial trace · See more »

Positive-definite matrix

In linear algebra, a symmetric real matrix M is said to be positive definite if the scalar z^Mz is strictly positive for every non-zero column vector z of n real numbers.

New!!: Trace (linear algebra) and Positive-definite matrix · See more »

Projection matrix

In statistics, the projection matrix \mathbf, sometimes also called the influence matrix or hat matrix \mathbf, maps the vector of response values (dependent variable values) to the vector of fitted values (or predicted values).

New!!: Trace (linear algebra) and Projection matrix · See more »

Quadratic form (statistics)

In multivariate statistics, if \varepsilon is a vector of n random variables, and \Lambda is an n-dimensional symmetric matrix, then the scalar quantity \varepsilon^T\Lambda\varepsilon is known as a quadratic form in \varepsilon.

New!!: Trace (linear algebra) and Quadratic form (statistics) · See more »

Rank (linear algebra)

In linear algebra, the rank of a matrix A is the dimension of the vector space generated (or spanned) by its columns.

New!!: Trace (linear algebra) and Rank (linear algebra) · See more »

Real number

In mathematics, a real number is a value of a continuous quantity that can represent a distance along a line.

New!!: Trace (linear algebra) and Real number · See more »

Representation theory

Representation theory is a branch of mathematics that studies abstract algebraic structures by representing their elements as linear transformations of vector spaces, and studies modules over these abstract algebraic structures.

New!!: Trace (linear algebra) and Representation theory · See more »

Scalar (mathematics)

A scalar is an element of a field which is used to define a vector space.

New!!: Trace (linear algebra) and Scalar (mathematics) · See more »

Semisimple Lie algebra

In mathematics, a Lie algebra is semisimple if it is a direct sum of simple Lie algebras, i.e., non-abelian Lie algebras \mathfrak g whose only ideals are and \mathfrak g itself.

New!!: Trace (linear algebra) and Semisimple Lie algebra · See more »

Similarity invariance

In linear algebra, similarity invariance is a property exhibited by a function whose value is unchanged under similarities of its domain.

New!!: Trace (linear algebra) and Similarity invariance · See more »

Simple Lie group

In group theory, a simple Lie group is a connected non-abelian Lie group G which does not have nontrivial connected normal subgroups.

New!!: Trace (linear algebra) and Simple Lie group · See more »

Skew-symmetric matrix

In mathematics, particularly in linear algebra, a skew-symmetric (or antisymmetric or antimetric) matrix is a square matrix whose transpose equals its negative; that is, it satisfies the condition In terms of the entries of the matrix, if aij denotes the entry in the and; i.e.,, then the skew-symmetric condition is For example, the following matrix is skew-symmetric: 0 & 2 & -1 \\ -2 & 0 & -4 \\ 1 & 4 & 0\end.

New!!: Trace (linear algebra) and Skew-symmetric matrix · See more »

Solvable Lie algebra

In mathematics, a Lie algebra \mathfrak is solvable if its derived series terminates in the zero subalgebra.

New!!: Trace (linear algebra) and Solvable Lie algebra · See more »

Specht's theorem

In mathematics, Specht's theorem gives a necessary and sufficient condition for two matrices to be unitarily equivalent.

New!!: Trace (linear algebra) and Specht's theorem · See more »

Special linear group

In mathematics, the special linear group of degree n over a field F is the set of matrices with determinant 1, with the group operations of ordinary matrix multiplication and matrix inversion.

New!!: Trace (linear algebra) and Special linear group · See more »

Special linear Lie algebra

In mathematics, the special linear Lie algebra of order n (denoted \mathfrak_n(F) or \mathfrak(n, F)) is the Lie algebra of n \times n matrices with trace zero and with the Lie bracket.

New!!: Trace (linear algebra) and Special linear Lie algebra · See more »

Square matrix

In mathematics, a square matrix is a matrix with the same number of rows and columns.

New!!: Trace (linear algebra) and Square matrix · See more »

Statistics

Statistics is a branch of mathematics dealing with the collection, analysis, interpretation, presentation, and organization of data.

New!!: Trace (linear algebra) and Statistics · See more »

Superalgebra

In mathematics and theoretical physics, a superalgebra is a Z2-graded algebra.

New!!: Trace (linear algebra) and Superalgebra · See more »

Supertrace

In the theory of superalgebras, if A is a commutative superalgebra, V is a free right A-supermodule and T is an endomorphism from V to itself, then the supertrace of T, str(T) is defined by the following trace diagram: More concretely, if we write out T in block matrix form after the decomposition into even and odd subspaces as follows, then the supertrace Let us show that the supertrace does not depend on a basis.

New!!: Trace (linear algebra) and Supertrace · See more »

Symmetric matrix

In linear algebra, a symmetric matrix is a square matrix that is equal to its transpose.

New!!: Trace (linear algebra) and Symmetric matrix · See more »

Tensor contraction

In multilinear algebra, a tensor contraction is an operation on a tensor that arises from the natural pairing of a finite-dimensional vector space and its dual.

New!!: Trace (linear algebra) and Tensor contraction · See more »

Tensor product

In mathematics, the tensor product of two vector spaces and (over the same field) is itself a vector space, together with an operation of bilinear composition, denoted by, from ordered pairs in the Cartesian product into, in a way that generalizes the outer product.

New!!: Trace (linear algebra) and Tensor product · See more »

Trace class

In mathematics, a trace class operator is a compact operator for which a trace may be defined, such that the trace is finite and independent of the choice of basis.

New!!: Trace (linear algebra) and Trace class · See more »

Trace inequalities

In mathematics, there are many kinds of inequalities involving matrices and linear operators on Hilbert spaces.

New!!: Trace (linear algebra) and Trace inequalities · See more »

Traced monoidal category

In category theory, a traced monoidal category is a category with some extra structure which gives a reasonable notion of feedback.

New!!: Trace (linear algebra) and Traced monoidal category · See more »

Transpose

In linear algebra, the transpose of a matrix is an operator which flips a matrix over its diagonal, that is it switches the row and column indices of the matrix by producing another matrix denoted as AT (also written A′, Atr, tA or At).

New!!: Trace (linear algebra) and Transpose · See more »

Triangular matrix

In the mathematical discipline of linear algebra, a triangular matrix is a special kind of square matrix.

New!!: Trace (linear algebra) and Triangular matrix · See more »

Vector space

A vector space (also called a linear space) is a collection of objects called vectors, which may be added together and multiplied ("scaled") by numbers, called scalars.

New!!: Trace (linear algebra) and Vector space · See more »

Vectorization (mathematics)

In mathematics, especially in linear algebra and matrix theory, the vectorization of a matrix is a linear transformation which converts the matrix into a column vector.

New!!: Trace (linear algebra) and Vectorization (mathematics) · See more »

Volume

Volume is the quantity of three-dimensional space enclosed by a closed surface, for example, the space that a substance (solid, liquid, gas, or plasma) or shape occupies or contains.

New!!: Trace (linear algebra) and Volume · See more »

0

0 (zero) is both a number and the numerical digit used to represent that number in numerals.

New!!: Trace (linear algebra) and 0 · See more »

Redirects here:

Exponential trace, Matrix trace, Trace (mathematics), Trace (matrix), Trace function, Trace matrix, Trace of a matrix, Trace of a square matrix, Trace of an endomorphism, Trace-free, Traceless.

References

[1] https://en.wikipedia.org/wiki/Trace_(linear_algebra)

OutgoingIncoming
Hey! We are on Facebook now! »