detexpkB

DetexpkB is a shorthand used in linear algebra and matrix analysis to denote the determinant of the matrix exponential exp(kB), where B is an n-by-n square matrix and k is a scalar parameter. The quantity is written as det(exp(kB)).

Because det(exp(A)) = exp(tr(A)) for any square matrix A, det(exp(kB)) = exp(tr(kB)) = exp(k tr(B)). This means that det(exp(kB)) depends only on the trace of B, and not on its full spectrum, beyond what is captured by the trace.

This leads to several consequences. If tr(B) = 0, then det(exp(kB)) = 1 for all k. If B is nilpotent, tr(B) = 0 as well, yielding the same result. The eigenvalues μ_i of B satisfy exp(kμ_i) as eigenvalues of exp(kB), so the determinant equals the product ∏ exp(kμ_i) = exp(k tr(B)) in agreement with the trace formula.

In applications, det(exp(kB)) can arise in stability analyses, differential equations, and control theory, where determinants of matrix exponentials appear in characteristic equations and related computations. The function det(exp(kB)) is an entire function of k, and its value is determined entirely by the trace of B.

Notes: det(exp(kB)) is a standard, compact way to encode the determinant of an exponential map of a matrix. It highlights the relationship between the exponential of a matrix, its eigenvalues, and the trace, via the identity det(exp(A)) = exp(tr(A)). See also matrix exponential, trace, and spectral mapping theorem.