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Let $\mathbf{\beta}$ be the parameter vector of a ridge regression.

Now we can say that:

\begin{equation} \frac{\partial \lambda \beta^T \beta}{\partial \beta}=2\lambda\beta. \end{equation}

Why is this?

I thought that $$\frac{d}{dx} x^t x = 2x^t$$

Which would imply that:

\begin{equation} \frac{\partial \lambda \beta^T \beta}{\partial \beta}=2\lambda\beta^{T}. \end{equation}

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Both are correct based on which layout notation (i.e. numerator or denominator) is being used. Unfortunately, this is not explicitly stated in most sources and you need to infer it.

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