Simultaneous reduction of a pair of quadratic forms

Mitra, Sujit Kumar ; Rao, C. Radhakrishna (1968) Simultaneous reduction of a pair of quadratic forms Sankhya - Series A, 30 (3). pp. 313-322. ISSN 0581-572X

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Given two quadratic forms Q1=x'Ax and Q2=x'Bx one of which (say Q2) is p.d., it is well known that both are simultaneously reducible to forms containing square terms only by a real non-singular transformation and also by contragradient transformations. In this paper necessary and sufficient conditions are obtained for other cases such as (a) Q1 arbitrary, Q2 n.n.d., (b) Q1 arbitrary, Q2 non singular, and (c) Q1 and Q2 are both arbitrary. For a pair of symmetric matrices A and B of the same order and B n.n.d., the paper introduces the concept of a proper eigen value and a proper eigen vector of A with respect to B. Such eigen vectors are shown to determine the required transformation for the simultaneous reduction of associated quadratic forms.

Item Type:Article
Source:Copyright of this article belongs to Indian Statistical Institute.
ID Code:32031
Deposited On:30 Mar 2011 12:53
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