Trace of a matrix

Properties

The trace is a linear map in the sense that

tr(A + B) = tr(A) + tr(B)     for all n-by-n matrices A and B

tr(rA) = r tr(A)        for all n-by-n matrices A and all scalars r.

A matrix and its transpose have the same trace:

tr(A) = tr(A^T).

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,

tr(ABC) = tr(CAB) = tr(BCA).

More generally, the same is true if the matrices are not assumed square, but are so shaped that all of these products exist.

If A and B are similar, i.e. if there exists an invertible matrix X such that A = X⁻¹BX, then by the cyclic property,

tr(A) = tr(B).

Because of this, one may define the trace of a linear map f : V -> V (where V is a finite-dimensional vector space) by choosing a basis for V, 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. This allows 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^*.

There exist matrices which have the same trace but are not similar.

If A is a square n-by-n matrix with real or complex entries and if λ₁,...,λ_n are the (complex) eigenvalues of A (listed according to their algebraic multiplicities), then

tr(A) = ∑ λ_i.

This follows from the fact that A is always similar to its Jordan form, an upper triangular matrix having λ₁,...,λ_n on the main diagonal.

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:

det(exp(A)) = exp(tr(A)).

The trace also prominently appears in Jacobi's formula for the derivative of the determinant (see under determinant).

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.

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 bounded linear operators on Hilbert spaces.

See also

trace class, field trace.