Multiplying Matrix X by X^T=2X^T?

Paula Cameron

Paula Cameron

Answered question

2022-11-07

Multiplying Matrix X by X T = 2 X T ?
In Elements of Statistical Learning, we differentiate R S S ( β ) = ( y X β ) T ( y X β ) w.r.t to β to get X T ( y X β )
According to some, this is because
( y X β ) T ( y X β ) = y T y 2 β T X T y + β T X T X β

Answer & Explanation

hitturn35

hitturn35

Beginner2022-11-08Added 20 answers

Noting that in this context, y is a N × 1 column vector, β is a ( p + 1 ) × 1 column vector, and X is an N × ( p + 1 ) matrix, so β T X T y is a scalar value, and as such is guaranteed to be symmetric regardless of X.
Hanna Webster

Hanna Webster

Beginner2022-11-09Added 3 answers

We should get β T X T y and y T X β. Note that this is a scalar.
Hence β T X T y = y T X β

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