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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 (or linear spaces), 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. ...
- tr(A) = A1,1 + A2,2 + ... + An,n.
where Aij represents the (i,j)'th element of A. The use of the term trace arises from the German term Spur (cognate with the English spoor). Cognates are words that have a common origin. ...
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. Scalar is a concept that has meaning in mathematics, physics, and computing. ...
Since the principal diagonal is not moved on transposition, a matrix and its transpose have the same trace: In mathematics, and in particular linear algebra, the transpose of a matrix is another matrix, produced by turning rows into columns and vice versa. ...
- tr(A) = tr(AT).
If A is an n×m matrix and B is an m×n matrix, then - tr(AB) = tr(BA).
Using this fact, 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 square matrices A, B, and C, A cyclic permutation is a permutation that shifts all elements of given ordered set by a fixed offset, with the elements shifted off the end inserted back at the beginning in the same order, i. ...
- tr(ABC) = tr(CAB) = tr(BCA).
More generally, the same is true if the matrices are not assumed to be square, but are so shaped such that all of these products exist.
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. Using the canonical isomorphism between the space End(V) of linear maps on V and V⊗V*, the trace of v⊗f is defined to be f(v), with v in V and f an element of the dual space V*. In mathematics, the dimension of a vector space V is the cardinality (i. ...
A vector space (or linear space) is the basic object of study in the branch of mathematics called linear algebra. ...
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 minimum set of vectors that, when combined, can address every vector in a given space. ...
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, the existence of a dual vector space reflects in an abstract way the relationship between row vectors (1×n) and column vectors (n×1). ...
Eigenvalue relationships If A is a square n-by-n matrix with real or complex entries and if λ1,...,λn are the (complex) eigenvalues of A (listed according to their algebraic multiplicities), then In mathematics, the real numbers are intuitively defined as numbers that are in one-to-one correspondence with the points on an infinite line—the number line. ...
In mathematics, the complex numbers are an extension of the real numbers by the inclusion of the imaginary unit i, satisfying . ...
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. ...
From the connection between the trace and the eigenvalues, one can derive a connection between the trace function, the matrix exponential function, and the determinant: In mathematics, the matrix exponential is a function on square matrices analogous to the ordinary exponential function. ...
In linear 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)).
The trace also prominently appears 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. ...
In mathematics, the derivative is one of the two central concepts of calculus. ...
In linear 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. ...
Other ideas and applications If one imagines that the matrix A describes a water flow, in the sense that for every x in Rn, the vector Ax represents the velocity of the water at the location x, then the trace of A can be interpreted as follows: given any region U in Rn, the net flow of water out of U is given by tr(A)· vol(U), where vol(U) is the volume of U. See divergence. A net flow network is a mere simplification notation over the standard positive flow network. ...
Volume, also called capacity, is a quantification of how much space an object occupies. ...
In vector calculus, the divergence is an operator that measures a vector fields tendency to originate from or converge upon a given point. ...
The trace is used to define characters of group representations. Given two representations A(x) and B(x), they are equivalent if tr A(x) = tr B(x). Representation theory is the branch of mathematics that studies properties of abstract groups via their representations as linear transformations of vector spaces. ...
A matrix whose trace is zero is said to be traceless or tracefree. 0 (zero), alternatively called naught or nought, is both a number and a numeral. ...
Inner product For an m-by-n matrix A with complex (or real) entries, we have - tr(A*A) ≥ 0
with equality only if A = 0. The assignment - <A, B> = tr(A*B)
yields an inner product on the space of all complex (or real) m-by-n matrices. // Definition Inner Product of two vectors Given twoN-by-1 column vectors v and u, the inner product is defined as the scalar quantity α resulting from where or equivalently indicates the conjugate transpose operator applied to vector v. ...
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. The word norm coming from the latin word norma which means angle measure or (lawlike) rule, has a number of meanings: A social or sociological norm; see norm (sociology). ...
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 bounded linear operators on Hilbert spaces. 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, 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...
In mathematics, a Hilbert space is an inner product space that is complete with respect to the norm defined by the inner product. ...
Partial trace is another generalization of the trace. In linear algebra and functional analysis, the partial trace is a generalization of the trace. ...
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