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Unformatted text preview: 1 . Use the Cauchy-Schwarz inequality to show that the arithmetic mean ( n-vector is greater than or equal to its harmonic mean. k xk )/n of a positive 10.2 Schur complements. Consider a matrix X = X T ∈ Rn×n partitioned as X= A BT B C , where A ∈ Rk×k . If det A = 0, the matrix S = C − B T A−1 B is called the Schur complement of A in X . Schur complements arise in many situations and appear in many important formulas and theorems. For example, we have det X = det A det S . (You don’t have to prove this.) (a) The Schur complement arises when you minimize a quadratic form over some of the variables. Let f (u, v ) = (u, v )T X (u, v ), where u ∈ Rk . Let g (v ) be the minimum value of f over u, i.e., g (v ) = inf u f (u, v ). Of course g (v ) can be −∞. Show that if A ≻ 0, we have g (v ) = v T Sv . (b) The Schur complement arises in several characterizations of positive definiteness or semidefiniteness of a block matrix. As examples we have the following three theorems: • X ≻ 0 if and only if A ≻ 0...
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This note was uploaded on 09/10/2013 for the course C 231 taught by Professor F.borrelli during the Fall '13 term at University of California, Berkeley.

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