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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., Linear algebra is the branch of mathematics concerned with the study of vectors, vector spaces (also called linear spaces), linear maps (also called linear transformations), and systems of linear equations. ...
For the square matrix section, see square matrix. ...
In linear algebra, the main diagonal of a square matrix is the diagonal which runs from the top left corner to the bottom right corner. ...
 where aij represents the entry on the ith row and ith column of A. Equivalently, the trace of a matrix is the sum of its eigenvalues, making it an invariant with respect to chosen basis. In mathematics, a number is called an eigenvalue of a matrix if there exists a nonzero vector such that the matrix times the vector is equal to the same vector multiplied by the eigenvalue. ...
In mathematics, in the fields of multilinear algebra and representation theory, invariants of tensors are coefficients of characteristic polynomial of the tensor A: The first invariant of an n×n tensor A () is the coefficient for (coefficient for is always 1), the second invariant ( is the coefficient for , etc. ...
The use of the term trace arises from the German term Spur (cognate with the English spoor), which, as a function in mathematics, is often abbreviated to "Sp". Look up cognate in Wiktionary, the free dictionary. ...
Properties
The trace is a linear map. That is, In mathematics, a linear transformation (also called linear operator or linear map) is a function between two vector spaces that respects the arithmetical operations addition and scalar multiplication defined on vector spaces, or, in other words, it preserves linear combinations. Definition and first consequences Formally, if V and W are...
- tr(A + B) = tr(A) + tr(B)
- tr(rA) = r tr(A)
for all square matrices A and B, and all scalars r. Note that the trace is only defined for a square matrix (i.e. n×n). In linear algebra, real numbers are called scalars and relate to vectors in a vector space through the operation of scalar multiplication, in which a vector can be multiplied by a number to produce another vector. ...
Since the principal diagonal is invariant under transposition, a matrix and its transpose have the same trace: In linear algebra, the transpose of a matrix A is another matrix AT (also written Atr, tA, or A′) created by any one of the following equivalent actions: write the rows of A as the columns of AT write the columns of A as the rows of AT reflect A...
- tr(A) = tr(AT).
If A is an m×n matrix and B is an n×m matrix, then both products AB and BA are square, and - tr(AB) = tr(BA).
We prove this by invoking the definition of matrix multiplication: In mathematics, matrix multiplication is the operation of multiplying a matrix with either a scalar or another matrix. ...
In particular, when both A and B are n by n, the trace vanishes on the derived algebra: tr([A,B]) = 0, and the trace gives a map of Lie algebras (where k is the scalar field, with the commutative Lie algebra structure). In mathematics, a Lie algebra is an algebraic structure whose main use is in studying geometric objects such as Lie groups and differentiable manifolds. ...
Using the commutativity of trace, we can deduce that the trace of a product of square matrices is equal to the trace of any cyclic permutation of the product, a fact known as the cyclic property of the trace. For example, with three matrices A, B, and C, shaped so that ABC, CAB, and BCA all exist,
- tr(ABC) = tr(CAB) = tr(BCA).
However, even if A, B, and C are square matrices of the same dimension, then the traces of their products does depend on the order of the product; i.e., not all permutations of the three letters are allowed. An example would be Then and .
For four or more matrices, any cyclic permutation is allowed; thus, for example, tr(ABCDE) = tr(EABCD).
However, if products of three symmetric matrices are considered, any permutation is allowed. (Proof: tr(ABC) = tr(AT BT CT) = tr((CBA)T) = tr(CBA).) For more than three factors this is not true. In linear algebra, a symmetric matrix is a matrix that is its own transpose. ...
The trace is similarity-invariant, which means that A and P−1AP (P invertible) have the same trace, though there exist matrices which have the same trace but are not similar. This can be verified using the cyclic property above: In mathematics, similarity invariance is a property exhibited by a function whose value is unchanged under similarities of its domain. ...
- tr(P−1AP) = tr(PP−1A) = tr(A)
Given some linear map f : V → V (V is a finite-dimensional vector space) generally, we can define the trace of this map by considering the trace of matrix representation of f, that is, choosing a basis for V and describing f as a matrix relative to this basis, and taking the trace of this square matrix. The result will not depend on the basis chosen, since different bases will give rise to similar matrices, allowing for the possibility of a basis independent definition for the trace of a linear map. In mathematics, the dimension of a vector space V is the cardinality (i. ...
In mathematics, a vector space (or linear space) is a collection of objects (called vectors) that, informally speaking, may be scaled and added. ...
Representation theory is the branch of mathematics that studies properties of abstract groups via their representations as linear transformations of vector spaces. ...
In linear algebra, a basis is a set of vectors that, in a linear combination, can represent every vector in a given vector space, and such that no element of the set can be represented as a linear combination of the others. ...
Such a definition can be given using the canonical isomorphism between the space End(V) of linear maps on V and V⊗V*, where V* is the dual space of V. Let v be in V and let f be in V*. Then the trace of the decomposable element v⊗f is defined to be f(v); the trace of a general element is defined by linearity. Using an explicit basis for V and the corresponding dual basis for V*, one can show that this gives the same definition of the trace as given above. In category theory, an abstract branch of mathematics, a natural transformation provides a way of transforming one functor into another while respecting the internal structure (i. ...
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. Dual vector spaces defined on finite-dimensional vector spaces can be used for defining tensors which are studied in tensor algebra. ...
If A and B are positive semi-definite matrices of the same order then: In mathematics, a definite bilinear form B is one for which B(v,v) has a fixed sign (positive or negative) when it is not 0. ...
Eigenvalue relationships If A is a square n-by-n matrix with real or complex entries and if λ1,...,λn are the (complex and distinct) eigenvalues of A (listed according to their algebraic multiplicities), then In mathematics, the real numbers may be described informally as numbers that can be given by an infinite decimal representation, such as 2. ...
A complex number can be visually represented as a pair of numbers forming a vector on a diagram called an Argand diagram In mathematics, the complex numbers are the extension of the real numbers obtained by adjoining an imaginary unit, denoted i, which satisfies:[1] Every complex number can be...
In mathematics, a number is called an eigenvalue of a matrix if there exists a nonzero vector such that the matrix times the vector is equal to the same vector multiplied by the eigenvalue. ...
In linear algebra, a scalar λ is called an eigenvalue (in some older texts, a characteristic value) of a linear mapping A if there exists a nonzero vector x such that Ax=λx. ...
- tr(A) = ∑ λi.
This follows from the fact that A is always similar to its Jordan form, an upper triangular matrix having λ1,...,λn on the main diagonal. In linear algebra, the Jordan normal form, also called the Jordan canonical form, named in honor of the 19th and early 20th-century French mathematician Camille Jordan, answers the question, for a given square matrix M over a field K, to what extent M can be simplified into a standard...
In the mathematical discipline of linear algebra, a triangular matrix is a special kind of square matrix where the entries below or above the main diagonal are zero. ...
Derivatives The trace is the derivative of the determinant: it is the Lie algebra analog of the (Lie group) map of the determinant. In mathematics, a Lie algebra is an algebraic structure whose main use is in studying geometric objects such as Lie groups and differentiable manifolds. ...
In mathematics, a Lie group, named after Norwegian mathematician Sophus Lie (IPA pronunciation: , sounds like Lee), is a group which is also a differentiable manifold, with the property that the group operations are compatible with the smooth structure. ...
This is made precise in Jacobi's formula for the derivative of the determinant (see under determinant). In matrix calculus, Jacobis formula expresses the differential of the determinant of a matrix A in terms of the adjugate of A and the differential of A. The formula is It is named after the mathematician C.G.J. Jacobi. ...
For other uses, see Derivative (disambiguation). ...
In algebra, a determinant is a function depending on n that associates a scalar, det(A), to every n×n square matrix A. The fundamental geometric meaning of a determinant is as the scale factor for volume when A is regarded as a linear transformation. ...
As a particular case,  : the trace is the derivative of the determinant at the identity.
From this (or from the connection between the trace and the eigenvalues), one can derive a connection between the trace function, the exponential map between a Lie algebra and its Lie group (or concretely, the matrix exponential function), and the determinant: In mathematics, the matrix exponential is a matrix function on square matrices analogous to the ordinary exponential function. ...
In algebra, a determinant is a function depending on n that associates a scalar, det(A), to every n×n square matrix A. The fundamental geometric meaning of a determinant is as the scale factor for volume when A is regarded as a linear transformation. ...
- det(exp(A)) = exp(tr(A)).
For example, consider the one-parameter family of linear transformations given by rotation through angle θ,  These transformations all have determinant 1, so they preserve area. The derivative of this family at θ = 0 is the antisymmetric matrix  which clearly has trace zero, indicating that this matrix represents an infinitesimal transformation which preserves area.
A related characterization of the trace applies to linear vector fields. Given a matrix A, define a vector field F on Rn by F(x) = Ax. The components of this vector field are all linear functions (given by the rows of A). The divergence div F is a constant function, whose value is equal to tr(A). By the divergence theorem, one can interpret this in terms of flows: if F(x) represents the velocity of a fluid at the location x, and U is a region in Rn, the net flow of the fluid out of U is given by tr(A)· vol(U), where vol(U) is the volume of U. For other uses, see Divergence (disambiguation). ...
In vector calculus, the divergence theorem, also known as Gausss theorem (Carl Friedrich Gauss), Ostrogradskys theorem (Mikhail Vasilievich Ostrogradsky), or Gauss-Ostrogradsky theorem 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...
A net flow network is a mere simplification notation over the standard positive flow network. ...
For other uses, see Volume (disambiguation). ...
The trace is a linear operator, hence its derivative is constant:
Applications The trace is used to define characters of group representations. Two representations, A(x) and B(x), are equivalent if tr A(x) = tr B(x). Group representation theory is the branch of mathematics that studies properties of abstract groups via their representations as linear transformations of vector spaces. ...
The trace also plays a central role in the distribution of quadratic forms. If is a vector of random variables, and is an -dimensional square matrix, then the scalar quantity is known as a quadratic form in . ...
Lie algebra A matrix whose trace is zero is said to be traceless or tracefree, and these matrices form the simple Lie algebra sln, which is the Lie algebra of the special linear group of matrices with determinant 1. The special linear group consists of the matrices which do not change volume, while the special linear algebra is the matrices which infinitesimally do not change volume. Zero redirects here. ...
In mathematics, a simple Lie group is a Lie group which is also a simple group. ...
In mathematics, a Lie algebra is an algebraic structure whose main use is in studying geometric objects such as Lie groups and differentiable manifolds. ...
In mathematics, the general linear group of degree n over a field F (such as R or C), written as GL(n, F), is the group of n×n invertible matrices with entries from F, with the group operation that of ordinary matrix multiplication. ...
Inner product For an m-by-n matrix A with complex (or real) entries and * being the conjugate transpose, we have - tr(A*A) ≥ 0
with equality only if A = 0. The assignment  yields an inner product on the space of all complex (or real) m-by-n matrices. In mathematics, an inner product space is a vector space with additional structure, an inner product (also called a scalar product), which allows us to introduce geometrical notions such as angles and lengths of vectors. ...
If m=n then the norm induced by the above inner product is called the Frobenius norm of a square matrix. Indeed it is simply the Euclidean norm if the matrix is considered as a vector of length n2. In linear algebra, functional analysis and related areas of mathematics, a norm is a function which assigns a positive length or size to all vectors in a vector space, other than the zero vector. ...
In mathematics, the term matrix norm can have two meanings: A vector norm on matrices, i. ...
In mathematics and astronomy, Euclidean space is a generalization of the 2- and 3-dimensional spaces studied by Euclid. ...
Generalization The concept of trace of a matrix is generalised to the trace class of compact operators on Hilbert spaces, and the analog of the Frobenius norm is called the Hilbert-Schmidt norm. In mathematics, a bounded linear operator A over a Hilbert space H is said to be in the trace class if for some (and hence all) orthonormal bases {ek}k of H the sum of positive terms is finite. ...
In functional analysis, a compact operator (or completely continuous 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 of Y. Such an operator is necessarily a...
The mathematical concept of a Hilbert space (named after the German mathematician David Hilbert) generalizes the notion of Euclidean space in a way that extends methods of vector algebra from the plane and three-dimensional space to spaces of functions. ...
In mathematics, a Hilbert-Schmidt operator is a bounded operator A on a Hilbert space H such that there exists an orthonormal basis of H with the property If this is true for one orthonormal basis, it is true for any other orthonormal basis. ...
The partial trace is another generalization of the trace that is operator-valued. In linear algebra and functional analysis, the partial trace is a generalization of the trace. ...
If A is a general associative algebra over a field k, then a trace on A is often defined to be any map tr: A → k which vanishes on commutators: tr([a,b]) = 0 for all a,b in A. Such a trace is not uniquely defined; it can always at least be modified by multiplication by a nonzero scalar. In mathematics, an associative algebra is a vector space (or more generally, a module) which also allows the multiplication of vectors in a distributive and associative manner. ...
A supertrace is the generalization of a trace to the setting of superalgebras. 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, Tr(T) is defined by the following tangle diagram: More concretely, if we write out T is block...
In mathematics and theoretical physics, a superalgebra over a field K is another name for a Z2-graded algebra over K. Specifically, a superalgebra is a super vector space A = A0 ⊕ A1 over K together with a bilinear multiplication which is an even morphism of super vector spaces. ...
The operation of tensor contraction generalizes the trace to arbitrary tensors. In multilinear algebra, a tensor contraction is a sum of products of scalar components of one or more tensors caused by applying the summation convention to a pair of dummy indices which are bound to each other in an expression. ...
See also In mathematics, a bounded linear operator A over a Hilbert space H is said to be in the trace class if for some (and hence all) orthonormal bases {ek}k of H the sum of positive terms is finite. ...
In mathematics, the field trace is a linear mapping defined for certain field extensions. ...
In mathematics, the Golden-Thompson inequality is as follows. ...
In probability theory, the characteristic function of any random variable completely defines its probability distribution. ...
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