Determinants and the volumes of parallelotopes and zonotopes

The Minkowski sum of edges corresponding to the column vectors of a matrix A with real entries is the same as the image of a unit cube under the linear transformation defined by A with respect to the standard bases. The geometric object obtained in this way is a zonotope, Z(A). If the columns of the matrix are linearly independent, the object is a parallelotope, P(A). In the first section, we derive formulas for the volume of P(A) in various ways as , as the square root of the sum of the squares of the maximal minors of A, and as the product of the lengths of the edges of P(A) times the square root of the determinant of the matrix of cosines of angles between pairs of edges. In the second section, we use the volume formulas to derive real-case versions of several well-known determinantal inequalities—those of Hadamard, Fischer, Koteljanskii, Fan, and Szasz—involving principal minors of a positive-definite Hermitian matrix. In the last section, we consider zonotopes, obtain a new proof of the decomposition of a zonotope into its generating parallelotopes, and obtain a volume formula for Z(A).