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I'm trying to solve the following derivative with respect to the scalar parameter $\sigma$

$$\frac{\partial}{\partial \sigma} \ln|\Sigma|,$$

where $\Sigma = (\sigma^2 \Lambda_K)$ and $\Lambda_K$ is the following symmetric tridiagonal $K \times K$ matrix $$ \Lambda_{K} = \left( \begin{array}{ccccc} 2 & -1 & 0 & \cdots & 0 \\ -1 & 2 & -1 & \cdots & 0 \\ 0 & -1 & \ddots & \ddots & \vdots \\ \vdots & \ddots & \ddots & \ddots & -1 \\ 0 & 0 & \ldots & -1 & 2 \\ \end{array}\right). $$

Is there a rule for these cases?

Thanks in advance for your time.

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2 Answers 2

up vote 3 down vote accepted

Edit (I didn't read the question carefully, here is the correct take): For any $K\times K$ matrix $\Lambda$, $\det(\sigma^2\Lambda) = \sigma^{2K}\det(\Lambda)$. Taking the log of the absolute value, we have $2K\log\sigma + \log|\det(\Lambda)|$, and then taking the derivative, the answer is $\frac{2K}{\sigma}$.

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Tks for pointing that out. It was extremely simple ... –  user60125 Jan 30 '13 at 23:13

Can anyone explain the more general case where Σ is a k by k matrix and its elements are function of the elements of vector b which in turn they are functions of a scalar σ?

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