The author’s point of departure is the following spezialized form of a theorem of B. Schwarz [Pac. J. Math. 32, 203–229 (1970; Zbl 0193.04501)]. Let \(Y(z)\) be any solution to the matrix differential equation (1) \(Y'(z) =AY(z)\) where \(A,Y\) are \(n\times n\) complex matrices. The \(p\)-th compound matrix \(C_p(Y(z))\) of \(Y(z)\), \(1\leq p\leq n\), satisfies the equation \(C_p(Y(z))'= B_p(A)C_p (Y(z))\). The \({n\choose p}\times{n\choose p}\) matrix \(B_p(A)\) is defined as follows: Let \(Q_{p,n}\) be the set of all strictly increasing sequences of \(p\) integers chosen from \(1,\dots,n\). The entries of \(B_p(A)=(b_{ij})\); \(i= (i_1,\dots,i_p), j=(j_1,\dots,j_p)\in Q_{p,n}\) with entries arranged lexicographically, are given by: \(b_{ij} = 0\), if at most \(p- 2\) of the indices of \(i\) coincide with indices of \(j\); \(b_{ij} = (-1)^{l+m}a_{i_\iota j_m}\) exactly \(p- 1\) of the indices of \(i\) coincide with indices of \(j\) and \(i_1,j_m\) is the remaining non coinciding pair; \(b_{ij}=\sum_{s=1}^pa_{i_sj_s}\) if \(i = j\).
In the present paper the author develops a number of results concerning the structure and eigenvalues of \(B_p(A)\) by using the connection with (1). For example, the eigenvalues of \(B_p(A)\) are all \({n\choose p}\) sums \(\sum_{s=1}^pa_{i_s}\) where \((i_1,\dots,i_p)\in Q_{p,n}\), and \(\alpha_1,\dots,\alpha_n\) are the eigenvalues of \(A\). Several applications of the characterizations of \(B_p(A)\) show that \(B_p(A)\) can be regarded as the additive analogue of the compound matrix \(C_p(A)\). While the compound matrix may be used to obtain inequalities for products of eigenvalues, \(B_p(A)\) may be used to obtain inequalities for sums of eigenvalues of \(A\). A corollary shows that \(B_p(A)\) is the first derivation associated with \(A\), operating on the \(p\)th Grassmann space [M. Marcus, Finite dimensional multilinear algebra (Pure and Applied Mathematics. Vol. 23. New York: Marcel Dekker) (1973; Zbl 0284.15024)].