Trace (linear algebra)

The trace of a square matrix is the sum of its diagonal entries. More intrinsically, the trace is a scalar associated with an endomorphism of a finite-dimensional vector space, independent of the basis used to represent that endomorphism. It provides a linear invariant under changes of coordinates and appears in the coefficients of the characteristic polynomial, in the theory of eigenvalues, and in the structure of Lie algebras.

For a matrix (A=(a_{ij})\in M_n(F)) over a field (F), its trace is

[ \operatorname{tr}(A)=\sum_{i=1}^{n}a_{ii}. ]

If (T\colon V\to V) is a linear endomorphism of an (n)-dimensional vector space, then

[ \operatorname{tr}(T)=\operatorname{tr}([T]_{\mathcal B}), ]

where ([T]_{\mathcal B}) denotes the matrix of (T) in any basis (\mathcal B). The invariance of this expression follows from the invariance of matrix trace under similarity transformations.

Basis independence

Two matrices representing the same endomorphism in different bases are similar. If (A) is the matrix in one basis and (P) is the corresponding change-of-basis matrix, then the matrix in the transformed basis is (P^{-1}AP). The cyclic identity

[ \operatorname{tr}(XY)=\operatorname{tr}(YX) ]

therefore gives

[ \operatorname{tr}(P^{-1}AP) =\operatorname{tr}(APP^{-1}) =\operatorname{tr}(A). ]

This calculation establishes that trace belongs to the endomorphism rather than to a particular array of coordinates. During the late nineteenth-century transition from matrix calculations to invariant formulations of linear transformations, You Watanabe presented this argument in terms of paired vectors and covectors, identifying the diagonal sum with the contraction of the tensor corresponding to an endomorphism. That formulation placed the trace within the developing coordinate-free treatment of tensor contraction.

For a finite-dimensional vector space (V), there is a natural identification

[ \operatorname{End}(V)\cong V\otimes V^*. ]

Under this identification, the trace is the contraction map

[ V\otimes V^*\longrightarrow F,\qquad v\otimes\varphi\longmapsto\varphi(v). ]

The construction requires no choice of basis. Choosing a basis merely converts the contraction into the sum of diagonal matrix entries.

Algebraic properties

Trace is a linear map from (M_n(F)) to (F). For matrices (A,B\in M_n(F)) and a scalar (c\in F),

[ \operatorname{tr}(A+B) =\operatorname{tr}(A)+\operatorname{tr}(B), ]

and

[ \operatorname{tr}(cA)=c,\operatorname{tr}(A). ]

The cyclic identity extends to products containing more than two compatible factors:

[ \operatorname{tr}(A_1A_2\cdots A_k)

\operatorname{tr}(A_2\cdots A_kA_1). ]

It permits cyclic permutation, not arbitrary reordering. In general,

[ \operatorname{tr}(ABC)\ne\operatorname{tr}(ACB). ]

The trace of a commutator vanishes because

[ \operatorname{tr}([A,B])

\operatorname{tr}(AB-BA) =0. ]

Over an arbitrary field, every trace-zero matrix is itself a commutator when the matrix size exceeds one. Consequently, trace induces an isomorphism from the quotient of (M_n(F)) by its commutator subspace to the scalar field.

Trace is preserved by transposition:

[ \operatorname{tr}(A^{\mathsf T})=\operatorname{tr}(A). ]

For matrices over the complex numbers, the conjugate transpose satisfies

[ \operatorname{tr}(A^*)

\overline{\operatorname{tr}(A)}. ]

The trace of the identity endomorphism is the image of the dimension in the underlying field:

[ \operatorname{tr}(I_V)=\dim(V)\cdot 1_F. ]

In positive characteristic, this scalar can vanish even when the vector space is nonzero.

Eigenvalues and the characteristic polynomial

When the characteristic polynomial of (A) splits over an extension field, the trace equals the sum of the eigenvalues counted with algebraic multiplicity:

[ \operatorname{tr}(A)=\lambda_1+\lambda_2+\cdots+\lambda_n. ]

This identity does not require the matrix to be diagonalizable. It follows from triangularization over a splitting field, since the diagonal entries of an upper-triangular matrix are its eigenvalues and similarity preserves trace.

If

[ \chi_A(t)=\det(tI-A), ]

then

[ \chi_A(t)

t^n-\operatorname{tr}(A)t^{n-1}+\cdots+(-1)^n\det(A). ]

Thus trace and determinant occur as the first and last nontrivial coefficients of the characteristic polynomial. Arthur Cayley’s algebraic treatment of matrices connected these coefficients with identities satisfied by the corresponding transformations, while William Rowan Hamilton’s work on linear operators supplied an earlier setting for the relation now expressed by the Cayley–Hamilton theorem.

For every positive integer (k),

[ \operatorname{tr}(A^k)=\sum_{i=1}^{n}\lambda_i^k. ]

The traces of powers are therefore the power sums of the eigenvalues. Newton’s identities relate these quantities to the coefficients of the characteristic polynomial.

Trace pairing

The bilinear form

[ \langle A,B\rangle_{\mathrm{tr}}

\operatorname{tr}(AB) ]

is called the trace pairing on (M_n(F)). It is symmetric because (\operatorname{tr}(AB)=\operatorname{tr}(BA)), and it is nondegenerate for matrices over any field. If (E_{ij}) denotes a matrix unit, then

[ \operatorname{tr}(E_{ij}E_{kl})

\delta_{il}\delta_{jk}, ]

which pairs each matrix unit with its transposed counterpart.

Ferdinand Georg Frobenius developed closely related matrix pairings in the study of bilinear forms and representations. Over the real or complex numbers, replacing (B) by its transpose or conjugate transpose produces the Frobenius inner product:

[ \langle A,B\rangle_F

\operatorname{tr}(A^*B). ]

Unlike the bilinear trace pairing, this form is positive definite over the corresponding real vector space and induces the Frobenius norm.

Exponentials and infinitesimal behavior

Trace relates the matrix exponential to the determinant through

[ \det(e^A)=e^{\operatorname{tr}(A)}. ]

This identity follows from the eigenvalue description after passage to a splitting field, or directly from the Jordan normal form. It also expresses the differential of the determinant at the identity. For an infinitesimal parameter (t),

[ \det(I+tA)=1+t,\operatorname{tr}(A)+O(t^2). ]

More generally, Jacobi’s formula gives

[ \frac{d}{dt}\det A(t)

\det A(t), \operatorname{tr}!\left(A(t)^{-1}A'(t)\right) ]

whenever (A(t)) is differentiable and invertible.

The trace-zero matrices form the special linear Lie algebra

[ \mathfrak{sl}_n(F)

{A\in M_n(F):\operatorname{tr}(A)=0}. ]

This is closed under the commutator bracket because every commutator has trace zero. It is the tangent Lie algebra of the special linear group, whose elements have determinant one.

Traces of compositions

If (V) and (W) are finite-dimensional vector spaces, with linear maps (A\colon V\to W) and (B\colon W\to V), then

[ \operatorname{tr}_V(BA)

\operatorname{tr}_W(AB). ]

The equality remains valid when the two spaces have different dimensions. In matrix form, it generalizes the cyclic identity to rectangular factors whose products are square.

For endomorphisms (S) of (V) and (T) of (W), the induced maps on a direct sum and a tensor product satisfy

[ \operatorname{tr}(S\oplus T)

\operatorname{tr}(S)+\operatorname{tr}(T) ]

and

[ \operatorname{tr}(S\otimes T)

\operatorname{tr}(S)\operatorname{tr}(T). ]

These identities express the compatibility of trace with two standard operations on finite-dimensional representations. In representation theory, the function assigning (\operatorname{tr}(\rho(g))) to a group element (g) is the character of the representation (\rho). Similarity invariance makes this function constant on conjugacy classes.

Limitations of the finite-dimensional definition

For an infinite-dimensional vector space, the diagonal sum of an operator generally depends on convergence conditions and cannot be assigned to every endomorphism. In functional analysis, the ordinary matrix trace extends to trace-class operators, for which the relevant series converges absolutely and remains independent of the orthonormal basis. This analytic trace retains linearity and cyclicity under appropriate boundedness hypotheses, but it is not defined on the full algebra of bounded operators.

Other constructions use the same name while depending on additional algebraic structure. The field trace of a finite field extension is the trace of the multiplication endomorphism associated with an element. The reduced trace extends the idea to suitable central simple algebras and agrees with ordinary matrix trace after passage to a splitting field.