On some Derivatives of Vector-Matrix Products Useful for Statistics Volume 1 - Issue 2

Michele Nichelatti*

• Service of Biostatistics, Fondazione Malattie del Sangue, Milan, Italy

Received: December 17, 2018;   Published: December 20, 2018

Corresponding author: Michele Nichelatti, Service of Biostatistics, Fondazione Malattie del Sangue, Milan, Italy

DOI: 10.32474/CTBB.2018.01.000110

Abstract PDF

In this brief description, we will use the numerator layout [1], and will tacitly assume that all products are conformable.

The derivative of the linear form 𝒰^t𝒱 with respect to the vector 𝒱is given as

and since 𝒰^t𝒱 is a scalar, we are facing a particular case of the derivative of a scalar λ with respect to a vector, e.g., ∂_𝒱λ=(∂_𝒱1λ ..... ∂_𝒱nλ) and it must also be ∂_𝒱(𝒰^t𝒱)=∂_𝒱(𝒱^t𝒰) . Moreover, it is easy to demonstrate that using the denominator layout, the derivative would have been ∂_𝒱(𝒰^t𝒱)=𝒰

If both 𝒰 and 𝒱 vectors are function of a third vector 𝒵, we get

which, in the case 𝒰=𝒱=𝒲 reduces to

Dealing with a linear transform 𝒰=a𝒱, if A is 𝓂×𝓂 we have

and if 𝒱 is a function of a vector 𝒲_𝒱 we get

From definition of bilinear form, we obtain, for 𝒰^tA𝒱 the derivative

while, for a quadratic form 𝒰^tA𝒱(where A is 𝓃 × 𝓃), we get

so that, if A is a symmetric matrix, say, for A = X^tX, then

1. Lütkepohl H (1997) Handbook of Matrices. John Wiley & Sons, New York, USA, pp. 320.