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.,
- <math>\mathrm{tr}(A) = a_{11} + a_{22} + \dots + a_{nn}=\sum_i a_{i i} \,</math>
where aij represents the entry on the ith row and jth column of A. Equivalently, the trace of a matrix is the sum of its eigenvalues, making it an invariant with respect to chosen basis. 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".
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Properties
The trace is a linear map. That is,
- 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). Since the principal diagonal is invariant under transposition, a matrix and its transpose have the same trace:
- tr(A) = tr(AT).
If A is an n×n matrix and B is an n×n matrix, then
- tr(AB) = tr(BA).
We prove this by invoking the definition of matrix multiplication:
- <math>\mathrm{tr}(AB) = \sum_{i=1}^n (AB)_{ii} = \sum_{i=1}^n \sum_{j=1}^n A_{ij} B_{ji} = \sum_{j=1}^n \sum_{i=1}^n B_{ji} A_{ij} = \sum_{j=1}^n (BA)_{jj} = \mathrm{tr}(BA).</math>
Thus the trace vanishes on the derived algebra: tr([A,B]) = 0, and the trace gives a map of Lie algebras <math>\mathfrak{gl}_{n+1} \to k</math> (where k is the scalar field, with the commutative Lie algebra structure). 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 square matrices A, B, and C,
- 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. If A, B, and C are square matrices of the same dimension and are symmetric, then the traces of their products are invariant not only under cyclic permutations but under all permutations, i.e.,
- tr(ABC) = tr(CAB) = tr(BCA) = tr(BAC) = tr(CBA) = tr(ACB).
The generalization of this identity to four or more symmetric matrices is not true however (the relevant products are not defined, in general). Any cyclic permutation is allowed; thus, for example, <math>tr(ABCDE)=tr(EABCD)</math>. 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:
- 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*. If A and B are positive semi-definite matrices of the same order then:
- <math> 0 \leq \mathrm{tr}(AB)^n \leq \mathrm{tr}(A)^n \mathrm{tr}(B)^n</math>
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
- 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.
Derivatives
The trace is the derivative of the determinant: it is the Lie algebra analog of the (Lie group) map of the determinant. This is made precise in Jacobi's formula for the derivative of the determinant (see under determinant). As a particular case, <math>\operatorname{tr}=\operatorname{det}'_I</math>: the trace is the derivative of the determinant at the identity. One can interpret this as a flow: 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. 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:
- det(exp(A)) = exp(tr(A)).
The trace is a linear operator, hence its derivative is constant:
- <math> {\rm d} {\rm Tr} ( {\mathbf X} ) = {\rm Tr}({\rm d} {\mathbf X})</math>
Applications
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). The trace also plays a central role in the distribution of quadratic forms.
Lie algebra
A matrix whose trace is zero is said to be traceless or tracefree, and these matrices form the simple Lie algebra sln+1, which is the Lie algebra of the special linear group of matrices with determinant 1: the special linear group is the matrices which don't change volume, while the special linear algebra is the matrices which infinitesimally don't change volume.
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
- <math>\langle A, B\rangle = \operatorname{tr}(A^*B)</math>
yields an inner product on the space of all complex (or real) m-by-n matrices. 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.
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. Partial trace is another generalization of the trace that is operator-valued.
