Ordinary least squares

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