detexpBA

detexpBA is a notational shorthand sometimes used to denote the scalar determinant of the matrix exponential of a product BA, i.e., det(exp(BA)). It arises in linear algebra and matrix analysis contexts where traces, determinants and exponentials of matrices are combined. The notation emphasizes the role of the ordered product BA rather than the individual factors.

Key property

A fundamental identity links determinants of matrix exponentials to traces: for any square matrix X over C or R, det(exp(X)) = exp(tr(X)). Because tr(BA) = tr(AB) when the products are defined, det(exp(BA)) = exp(tr(BA)) = exp(tr(AB)) = det(exp(AB)). This reduces evaluation of det(exp(BA)) to computing a trace rather than a full matrix exponential and determinant.

Domain and generalizations

detexpBA is defined when BA is a square matrix (for instance when A is n×m and B is m×n, BA is n×n). The identity carries over to complex matrices and is widely used in contexts involving rectangular factor pairs through the equality of traces.

Applications and computation

The trace-based reduction makes det(exp(BA)) useful in control theory, statistical mechanics, and spectral analysis, where exponential determinants appear in partition functions, characteristic measures, and stability criteria. Numerically, computing tr(BA) is cheaper and often more stable than computing exp(BA) followed by its determinant, though round-off and conditioning should be considered.