det-trace identities

Determinant and trace are the two simplest basis-free numbers attached to an operator, and a web of identities ties them together. The headline is det(exp A) = exp(tr A): exponentiating a matrix turns the trace (a sum) into the determinant (a product) through the exponential. The two invariants are not independent curiosities; they are the extreme coefficients of one polynomial, the characteristic polynomial.

Both are elementary symmetric functions of the eigenvalues lambda_1, ..., lambda_n. The determinant is their PRODUCT, det = lambda_1 * ... * lambda_n, and the trace is their SUM, tr = lambda_1 + ... + lambda_n. More fully, the characteristic polynomial det(t*I - A) = t^n - (tr A) t^(n-1) + ... + (-1)^n det(A) has, as its coefficients, the elementary symmetric functions e_1, ..., e_n of the eigenvalues; trace is e_1 (up to sign) and determinant is e_n. Every coefficient in between is another symmetric function, a 'higher trace'.

The identity det(exp A) = exp(tr A) follows cleanly from this: the eigenvalues of exp A are exp(lambda_i), so det(exp A) = product of exp(lambda_i) = exp(sum of lambda_i) = exp(tr A). This is the linear-algebra heart of Liouville's theorem in dynamics (volume growth of a flow is governed by the trace of its generator) and of the formula d/dt det(M(t)) = det(M) * tr(M^-1 M') for a moving matrix.

Caveat: tr A is the sum and det A the product of eigenvalues counted WITH algebraic multiplicity, and over the complex numbers (or any algebraically closed field) so that all eigenvalues exist. Over the reals a matrix may lack a full set of real eigenvalues, yet trace and determinant remain real and well defined as polynomial-in-the-entries quantities — they are honest invariants regardless, just no longer literally a real product or sum of real eigenvalues.