matrix_facts_2009_11_18_01_2up

# matrix_facts_2009_11_18_01_2up - 16 - 1 Matrix facts S....

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16 - 1 Matrix facts S. Lall, Stanford 2009.11.18.01 16 - Matrix facts Completion of squares Block LDU matrix decomposition Inverse of a block matrix Inverse of a sum Useful matrix identities Push-through identity 16 - 2 Matrix facts S. Lall, Stanford 2009.11.18.01 Completion of Squares the completion of squares formula for quadratic polynomials is ax 2 + 2 bxy + dy 2 = a ± x + b a y ² 2 + ± d - b 2 a ² y 2 when a > 0 , this tells us the minimum with respect to x for Fxed y min x R ax 2 + 2 bxy + dy 2 = ± d - b 2 a ² y 2 which is achieved when x = - b a y . this also gives a test for global positivity : ax 2 + 2 bxy + dy 2 > 0 for all nonzero x, y R ⇐⇒ a > 0 and d - b 2 a > 0

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16 - 3 Matrix facts S. Lall, Stanford 2009.11.18.01 completion of squares for matrices if A R n × n and D R m × m are symmetric matrices and B R n × m , then ± x y ² T ± A B B T D ² ± x y ² = ( x + A - 1 By ) T A ( x + A - 1 By ) + y T ( D - B T A - 1 B ) y compare with
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## This note was uploaded on 08/23/2010 for the course EE 263 taught by Professor Boyd,s during the Fall '08 term at Stanford.

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matrix_facts_2009_11_18_01_2up - 16 - 1 Matrix facts S....

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