XXII.—Note on a Theorem of Frobenius' connected with Invariant-Factors

Extract

The theorem in question appears in his well-known paper of the year 1894, “Ueber die Elementartheiler der Determinanten.” Save that English takes the place of German, the enunciation of it stands exactly as follows:—

If ξ ≦ r ≦ s < n, and D_ξbe any determinant of the n^th order in the array

D_ξ′ being its complementary minor of the (n − ξ)^th order in the determinant of the n^th order

then