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I am reading Boyd's Convex Optimization Text, and I am curious to know why the following is true:

$$F F^T \succeq F^* {F^*}^T,$$

where $F^* {F^*}^T = (A^T A)^{-1}$ and $FA = I.$

I already tried multiplying both sides by $(A^T A)$, but could not get $AF$ to simplify. The context here is finding the minimum error of an unbiased estimator. The page number is 177, line 11.


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up vote 2 down vote accepted

We are trying to minimize $\mathrm E(\hat x - x)(\hat x-x)^T = F F^T$ subject to $FA = I$, where $A$ is the matrix in $y=Ax + v$ relating the measurements to the states, and the estimate is $\hat x = F y$.

It can be seen that $F$ is a pseudo-inverse of $A$. Call the optimum solution $F^*$. The claim is that $F^* = A^\dagger = (A^T A)^{-1} A^T$. I assume you do not want proof of this.

$F F^T \succeq F^*F^{*^T}$ follows from our designation of $F^*$ as the optimal solution. On the previous page it states

... the first estimator is at least as good as the second, i.e., $F_1 F_1^T \preceq F_2F_2^T$, if and only if ...

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