detexpY

DetexpY is a compact notation sometimes used to denote the determinant of the matrix exponential of a square matrix Y, written as det(exp(Y)) or det(e^Y). Here Y is an n-by-n matrix over the real or complex numbers.

Definition and basic property. For a square matrix Y, the determinant of its exponential is det(exp(Y)). A fundamental identity is det(exp(Y)) = exp(tr(Y)), where tr(Y) is the trace of Y. This follows from the eigenvalues λ_i of Y, since the eigenvalues of exp(Y) are e^{λ_i} and det(exp(Y)) = ∏ e^{λ_i} = e^{∑ λ_i} = e^{tr(Y)}. Consequently, det(exp(Y)) is always a positive real number when Y is real.

Key implications. The quantity det(exp(A+B)) = det(exp(A)) det(exp(B)) holds for all A,B because tr(A+B) = tr(A) + tr(B). However, exp(A+B) generally differs from exp(A) exp(B) unless A and B commute; only the determinants align via the trace identity. Numerically, det(exp(Y)) can be computed directly as exp(tr(Y)). In ill-conditioned or large-scale settings, using the log-det form, log det(exp(Y)) = tr(Y), helps avoid overflow.

Examples and applications. If Y = [ [a, 0], [0, d] ], then det(exp(Y)) = exp(a+d). In statistics and machine learning, det(exp(Y)) relates to log-determinant terms in Gaussian models. In differential geometry and Lie theory, it reflects volume changes under exponential maps from the Lie algebra to GL(n).

Note. The term detexpY is not universal; some texts simply write det(exp(Y)) or exp(tr(Y)).