An algebraic proof of the law of inertia starts with the orthonormal eigenvectors

Chapter 6, Problem 6.2.36

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An algebraic proof of the law of inertia starts with the orthonormal eigenvectors x1,...,xp of A corresponding to eigenvalues i > 0. and the orthonormal eigenvectors y1,...,yq of C TAC corresponding to eigenvalues i < 0. Positive Definite Matrices (a) To prove that the p+q vectors x1,...,xp, Cy1,...,Cyq are independent, assume that some combination gives zero: a1x1 ++apxp = b1Cy1 ++bqCyq (= z, say). Show that z TAz = 1a 2 1 ++pa 2 p 0 and z TAz = 1b 2 1 ++ qb 2 q 0. (b) Deduce that the as and bs are zero (proving linear independence). From that deduce p+q n. (c) The same argument for the n p negative s and the n q positive s gives n p + n q n. (We again assume no zero eigenvalueswhich are handled separately). Show that p + q = n, so the number p of positive s equals the number nq of positive swhich is the law of inertia.