Roughly calculation of ordinary least squares of multiple linear regression.

y = X b + e

is a multiple linear regression where

y is a dependent vector,

X is an explanatory matrix and

e is an error vector.

e = y - X b

Multiply both sides with a transposed variable of then self.

eT e

= (y - X b)T (y - X b)

= yT y - 2 bT XT y + bT XT X b

Differentiate by b.

Since the right hand side of previous equation is a squared expression with b,

the differentiation of minimal value of it is 0.

d(eT e) / db = - 2 XT y + 2 XT X b = 0

b = (XT X)^-1 XT y

References:

https://en.wikipedia.org/wiki/Ordinary_least_squares

https://en.wikipedia.org/wiki/Proofs_involving_ordinary_least_squares#Least_squares_estimator_for_.CE.B2

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