matrix_facts_2009_11_18_01

matrix_facts_2009_11_18_01 - 16 1 Matrix facts S Lall...

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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
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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 parenleftbigg x + b a y parenrightbigg 2 + parenleftbigg d - b 2 a parenrightbigg y 2 when a > 0 , this tells us the minimum with respect to x for fixed y min x R ax 2 + 2 bxy + dy 2 = parenleftbigg d - b 2 a parenrightbigg 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 bracketleftbigg x y bracketrightbigg T bracketleftbigg A B B T D bracketrightbiggbracketleftbigg x y bracketrightbigg = ( x + A - 1 By ) T A ( x + A - 1 By ) + y T
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