# General Solution to $\operatorname{Tr} \ln ( I + A)$ where A is complex and symmetric and zero on the diagonal

Are there any useful identities which would help me to find a general formula for

$\operatorname{Tr} \ln ( I + A )$

Where I is the identity matrix and A is some N by N complex and symmetric matrix, which has zeroes along the diagonal. I have tried splitting A up into upper and lower triangular components and expanding the log with the mercator series but it's starting to get pretty messy already so I think I missed something. Another property of A which could be useful is that it has a special repeating form along the rows and columns:

\begin{array}{ccccccc} 0 & b & c & d & e ..\\ b & 0 & b & c & d.. \\ c & b & 0 & b & c...\\ d & c & b & 0 & b...\\ . & . & . & . & ...\ \end{array}

It's been a while since I studied matrices so if there is a special name for this I've forgotten!

EDIT: Did a bit of research, this type of matrix is a "Symmetric Toeplitz Matrix".

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The only thing I could think of is an identity $\mathrm{Tr}\ln(1+A)=\ln\mathrm{det}(1+A)$. – Start wearing purple May 21 '13 at 17:21
@O.L., your identity becomes problematic of $I+A$ has any negative eigenvalues. – Michael Grant May 21 '13 at 18:58
The go-to reference here is going to be Functions of Matrices. Theory and Computation by Nicholas Higham. Fortunately, online sources reproduce some useful pieces of information from this book, because I don't have a copy. – Michael Grant May 21 '13 at 19:01
@MichaelC.Grant: Here there is a conservation of problems in going from the left to the right. One will have the same difficulties when defining $\ln (1+A)$. – Start wearing purple May 21 '13 at 19:17
That's fair. The logarithm exists if $I+A$ is nonsingular, but to define a unique principal logarithm you need for $I+A$ to have no negative real eigenvalues. – Michael Grant May 21 '13 at 19:38