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Homework Help: Schur complement question

  1. Apr 8, 2010 #1
    Let M be a real symmetric and positive definite matrix with blocks A, Bt, B and C.

    [tex] M= [[A,B^{t}] ; [B,C]] [/tex]

    where [tex] A[/tex] is a [tex] p\times p[/tex] matrix;[tex] B[/tex] is [tex] q\times p[/tex]; and [tex] C[/tex] is [tex] q\times q[/tex].

    Let [tex] S=C-BA^{-1}B^{t}[/tex] be the Schur complement. We prove that S is symmetric positive definite.

    I can prove that S is symmetric but I am having trouble proving that it is positive definite.
    I know that for S a symmetric matrix, S positive definite is equivalent to say that all eigen values of S are positive.

    I guess my question is how do we prove that all eigen values of S are positive?
  2. jcsd
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