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mihir_naik

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  1. Dear all, I have a query regarding the Sylvester's Law of Inertia (Congruence Transformation). It says the following (taken from Gilbert Strang---- 4th edition, Page 324) : C^T*A*C has the same number of positive, negative and zero eigenvalues as A, for some nonsingular matrix C. That is, the signs of eigenvalues are preserved by the congruence transformation. Here, I give 2 matrices A and C^T*A*C where the signs of the eigenvalues do not seem to be preserved: A = -10.0000 -0.2000 -0.1500 5.0000 0.1440 0.3600 5.0000 -2.0000 0 -2.5000 1.4400 0 0 1.0000 -3.0000 0 -0.7200 7.2000 -5.0000 0 0 2.5000 0 0 0 -1.0000 0 0 0.7200 0 0 0 -1.5000 0 0 3.6000 Eigenvalues of A = -7.3508, -1.3815, 0, 0, 0, 0.5524 (Note that A is a singular matrix and hence it has zero eigenvalues) C = 0.3162 0 0 0 0 0 0 0.7071 0 0 0 0 0 0 0.5774 0 0 0 0 0 0 0.4472 0 0 0 0 0 0 0.8333 0 0 0 0 0 0 0.3727 C^T*A*C = -0.66332, -0.0000094371,0, 0.0002162, -0.4184-0.2645*i, -0.4184 + 0.2645*i It is to be noted that C^T*A*C has two complex conjugate eigenvalues (and because A is singular, C^TAC is also singular), whereas A has only real eigenvalues. But it can be seen that A has two negative eigenvalues whereas C^T*A*C seems to have 4 negative eigenvalues (including the complex conjugates). Any help on this issue is appreciated. Thanks.
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