2. The index (3.1.8) becomes E(W, b) = =1 n i=1(yi (Wx )i bi )2, where yi is the ith component of

2. The index (3.1.8) becomes E(W,

b) = 

κ=1

n i=1(yκi − (Wxκ )i − bi )2, where yκi is the ith component of the κth target yκ . In order to find normal equations, define

ˆW

=

⎛

⎜⎜⎜⎜⎜⎝

w11 w12 . . . w1n w21 w22 . . . w2n

......

wd1 wd2 . . . wdn b1 b2 . . . bn

⎞

⎟⎟⎟⎟⎟⎠

, and let (Y )ij = yij with Y ∈ R

,n. Then E( ˆ W) = tr



(Y − ˆX

ˆW

)(Y

− ˆW

ˆX

)



=

tr(YY

) − 2 tr(Y ˆW

ˆX

) + tr( ˆX

ˆW

ˆW

ˆX

), and the normal equations are obtained imposing

∂ ˆW E = 0. Since ∂ ˆW tr(Y ˆW

ˆX

) = ˆX Y and ∂ ˆW tr( ˆX

ˆW

ˆW

ˆX

) = 2( ˆX

ˆX ˆ W),