Hall's theorem and compound matrices

Compound matrices and three celebrated theorems

“…The rank of AB can never be greater than the lesser of the ranks of the two individual matrices. It was shown in [2] that Hall's theorem can be derived in a few steps if we use compound matrices. However, our intention in this paper was to restrict ourselves to the three celebrated theorems.…”

“…Since, AB = I(4×4), (AB) (2) = I(6×6). By actual multiplication, we get, A (2) B (2) = I(6 × 6). Since, AB = I(4 × 4), (AB) 2 = I(6 × 6).…”

On the confluent Vandermonde matrix calculation algorithm

“… Spectral problems of ordinary and partial differential equations: Linsay, Rooney (1992) [43].  Graph theory: Article Nambiar (1997) [58] used compound matrices to give a concise and elegant proof of Hall's Theorem.  Article Prells, Friswell (2003) [61] successfully tackled the classic hard problem of matrix algebra, i.e.…”

Use of geometric algebra: compound matrices and the determinant of the sum of two matrices

“…A natural generalization of determinants are compound matrices which are closely related to the presentation of geometric algebra. As with geometric algebra, the concept of compound matrices has been neglected for many years, and its potential in theoretical analysis and practical application has been widely underestimated (see, for example, Nambiar 1997;Nambiar & Keating 1970;Malik 1970;Linsay & Rooney 1992;Fuchs 1992;Mitrouli & Koukouvinos 1997). For example, Nambiar & Keating (1970) conclude their paper by saying while a few network theorists certainly seem to be aware of the potentialities here, largely compound matrices seem to be a neglected subject.…”