Monday, 7 March 2022

Lupine Publishers| On some Derivatives of Vector-Matrix Products Useful for Statistics

 Lupine Publishers| Journal of Biostatistics & Biometrics

Mini Review

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 𝒰t𝒱 with respect to the vector 𝒱is given as

and since 𝒰t𝒱 is a scalar, we are facing a particular case of the derivative of a scalar Ξ» with respect to a vector, e.g., ∂𝒱λ=(∂𝒱1Ξ» ..... ∂𝒱nΞ») and it must also be ∂𝒱(𝒰t𝒱)=∂𝒱(𝒱t𝒰) . Moreover, it is easy to demonstrate that using the denominator layout, the derivative would have been ∂𝒱(𝒰t𝒱)=𝒰

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

From definition of bilinear form, we obtain, for 𝒰tA𝒱 the derivative

while, for a quadratic form 𝒰tA𝒱(where A is 𝓃 × π“ƒ), we get

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