In the post that derives the least squares estimator, we make use of the following statement:

This post shows how one can prove this statement. Let’s start from the statement that we want to prove:

Note that is symmetric. Hence, in order to simplify the math we are going to label as A, i.e. .

Let’s compute the partial derivative of with respect to .

Instead of stating every single equation, one can state the same using the more compact matrix notation:

plugging in for A

Now let’s return to the derivation of the least squares estimator.

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