In this paper we have shown that the existence of some balanced incomplete block (b.i.b. for abbreviation) designs implies the existence of some others in some cases. We have here established the following theorems.Theorem 1. If there exists a symmetric b.i.b. design with parametersv, b, r, k, λ, (where v-b, r=k), then there exists a b.i.b. design with parameters,
In this paper some methods for the construction of balanced incomplete block (b.i.b. for conciseness) designs are given. In the last section it is established that the existence of an affine resolvable b.i.b. design implies the existence of two other b.i.b. designs; §§ 6 and 7 are independent of §§ 3, 4, and 5.
All our matrices are square with real elements. The Schur product of two n × n matrices B = (bij) and C = (cij) (i, j, = 1, 2, …, n), is an n × n matrix A = (aij) with aij = bij cij, (i, j = 1, 2, …, n).A result due to Schur [1] states that if B and C are symmetric positive definite matrices then so is their Schur product A. A question now a rises. Can any symmetric positive definite matrix be expressed as a Schur product of two symmetric positive definite matrices? The answer is in the affirmative as we show in the following theorem.
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