detabcd

Detabcd is a term that appears in some linear algebra discussions to denote the determinant of a 2-by-2 block matrix composed of four equal-sized square blocks A, B, C, and D. In this usage, the block matrix is written as [A B; C D], and detabcd refers to the determinant of the entire 2n-by-2n matrix formed from these blocks. The notation is not universally standardized; many texts simply write det([A B; C D]) or det M for the corresponding block matrix.

A key property is that, if A is invertible, the determinant can be expressed using the Schur complement: det([A B; C D]) = det(A) det(D − C A^{-1} B). If D is invertible, an equivalent form is det([A B; C D]) = det(D) det(A − B D^{-1} C). In the special case where A, B, C, D are scalars (i.e., n = 1), detabcd reduces to ad − bc, the standard determinant formula for a 2×2 matrix.

Computation is typically performed by treating the block matrix as a whole and applying standard determinant algorithms, or by computing a Schur complement when a block is invertible. The concept is useful in areas that work with block-structured matrices, such as control theory, multivariate statistics (covariance matrices with block structure), and numerical linear algebra.

Limitations include the need for invertibility of a block to apply the Schur-complement form; otherwise, other determinant computation methods are used. Because detabcd is not a universally adopted notation, some authors prefer explicit block-matrix notation or direct determinant formulas without naming the block-determinant.