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Published online by Cambridge University Press: 20 January 2009
In a paper entitled “On differentiating a Matrix” H. W. Turnbull deduced some interesting and elegant results by the use of a matrix operator Ω, a matrix whose elements were the partial differential operators with respect to the elements of a square matrix x. Throughout the present paper the differential operator Ω. is used, or rather a matrix operator, which is the product of Ω and another square matrix Y.
By means of this operator in §§ 1 and 2 Bazin's matrix and Reiss's matrix are considered from the standpoint of matrices as distinct from that of determinants. Reiss's matrix is shown to be a constant times a compound of Bazin's matrix; and the latent roots of Reiss's matrix are immediately determined in terms of the latent roots of Bazin's matrix. From this result a theorem, discovered by Deruyts, is deduced as well as a more general theorem.
page 240 note 1 Turnbull, H. W., “On Differentiating a Matrix,” Proc. Edin. Math. Soc., Ser. 2, Vol. 1, Part 2, pp. 111–128 (1928). This paper will be referred to as Turnbull's paper.CrossRefGoogle Scholar
page 240 note 2 On Bazin, (1851) see Muir, “Theory of Determinants” II, 206–208, and on Reiss (1867) see Muir, III, 181, 189, and also on Picquet, see III, 198–199, and on Deruyts, IV, 15. The work of these authors was purely determinantal.Google Scholar
page 240 note 3 We define the matrix ' instead of the matrix X for convenience; in particular it conforms more closely with the notation of Turnbull's paper, which has already been cited.
page 241 note 1 Turnbull, . Loc. cit., page 112.Google Scholar
page 241 note 2 Throughout this paper repeated Greek suffixes will denote summation from 1 to n.
page 241 note 3 Throughout this paper we shall take the matrices X and y to be non-singular. If Δ = 0, the matrix X -1 Δ in (9) must be replaced by the adjugate matrix of X.
page 241 note 4 Bazin, . Loc. cit., page 148.Google Scholar
page 242 note 1 Reiss, M.. Loc. cit. It is important to notice that the relationship given by (14) holds between a minor of M and the complementary minor of N' . This fact is not made clear in the statement of the theorem ,in Turnbull's “Determinants, Matrices, and Invariants” page 109.Google Scholar
page 244 note 1 The elements of the two matrices R r and M r must naturally correspond ; i.e. if an element of R r is the determinant of the matrix obtained by replacing the k 1,......k r of X' by the columns s1, ....., sr of Y', then the corresponding element M r is the r-rowed minor of M formed from the rows s1, ...... sr and the columns k1, ......, kr of M.
page 244 note 2 Pascal, E., “Repertorium der höheren Mathematik,” I, Analysis, page 139.Google Scholar
page 245 note 1 Reiss, M.. Loc. cit. This first theorem is sometimes known as Picquet's theorem. See Muir, “Theory of Determinants,” Vol. III, page 198.Google Scholar
page 245 note 2 Muir, “Theory of Determinants,” IV, 215–217.
page 245 note 3 Cf. Muir, IV, 15.
page 249 note 1 This corollary is a special case of the Sylvester-Frobenius Theorem that if λ is a latent root of a matrix A, then f(λ.) is a latent root of the matrix polynomial f(A).
page 250 note 1 Turnbull, . Loc. cit. Formula (12) § 1 and formula (4) § 2.Google Scholar
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