In least squares estimation when $X^tX$ is not full rank, a solution can be found using the Moore Penrose pseudo Inverse.

The Moore Penrose pseudo inverse of $X^tX$ gives a solution of the linear system $X^tX a = X^t b$. For each b, $b_{mp}= (X^tX)^{-\perp} X^t b$ is the solution of miniminal $\| \|_2$ norm among all posible solutions.

Now consider$\| a \|^2_D = a^tDa$ with $D$ a positive definite matrix.

Is there a pseudo inverse $(X^tX)^D$ such that for every $b$, $b_D = (X^tX)^D X^t b$ is the solution of minimial $\|\|_D$ norm for the linear system?


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