Take the 2-minute tour ×
Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. It's 100% free, no registration required.

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.

share|improve this question
add comment

2 Answers

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 $\frac{2K}{\sigma}$.

share|improve this answer
    
Tks for pointing that out. It was extremely simple ... –  user60125 Jan 30 '13 at 23:13
add comment

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?

share|improve this answer
add comment

Your Answer

 
discard

By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.