Is the following true ?

theorem and proof

The inverse of a positive definite matrix is also positive definite and since symmetric then we could write the following:

$A=PP^T, \space B = A^{-1} = P\Lambda P^T$ since $AA^{-1}=I$ we could also write $AA^{-1}=(PP^T)(P\Lambda P^T) = P(P^TP)\Lambda P^T = PA^T\Lambda P^T = I$ and since $P$ is non singular $A^T\Lambda = P^{-1}P^{-T} = (P^T P)^{-1} = A^{-T}$ so after transposition $\Lambda A = A^{-1}$

The result does not seem correct, but I can't find mistakes in its derivation.

  • $\begingroup$ Where does your second equation $A^{-1} = P\Lambda P^T$ come from? That is $B$, isn't it? Or did I misread the theorem which I consider as two equations for $A$ and $B$ separately? $\endgroup$
    – Dr_Be
    Feb 8, 2016 at 9:43
  • $\begingroup$ Yes, $A^{-1}$ is symmetric so $B = A^{-1}$ (I added this to the question) $\endgroup$ Feb 8, 2016 at 9:49


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