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Let $Z$ be a 3$\times$2 matrix, $G$ be a 2$\times$2 positive definite symmetric (covariance) matrix, $\sigma^2$ a positive scalar, and $I$ a 2$\times$2 unit diagonal matrix.

Numerically I always see that

$$Z^{\prime}(ZGZ^{\prime}+\sigma^2I)^{-1}Z < (G^{-1}+Z^{\prime}Z/\sigma^2)$$

Is that true always ?

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  • $\begingroup$ Yes, it is always true. Actually the left hand side is less than either of the terms on the right hand side, let alone the sum of the two of them. $\endgroup$
    – Gordon Smyth
    Commented Nov 27, 2017 at 4:50
  • $\begingroup$ @GordonSmyth can you please provide a simple proof $\endgroup$
    – raK1
    Commented Nov 27, 2017 at 4:55

1 Answer 1

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For any nonzero 2-vector $x$, $$x'Z(ZGZ'+\sigma^2I)^{-1}Zx \le x'Z(\sigma^2I)^{-1}Zx=x'Z'Zx/\sigma^2< x'(G^{-1}+Z'Z/\sigma^2)x$$

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