The Matrix Cookbook

Stephen Wolfram, the guy behind the software Mathematica, has launched something called  Wolfram Alpha. Wolfram Alpha is a kind of search engine with a very ambitious goal; in their own words: “Wolfram|Alpha’s long-term goal is to make all systematic knowledge immediately computable and accessible to everyone. We aim to collect and curate all objective data; implement every known model, method, andalgorithm; and make it possible to compute whatever can be computed about anything.” Ambitious indeed. 

Typing in “Matrix”, the result is a help page on matrix computations, i.e., one can compute the trace, determinant, eigenvalue, etc about a specific matrix typed in, in a Mathematica-like syntax. The output for a specific matrix is a colorplot of the matrix along with the computed trace, determinant, eigenvalues, eigenvectors and inverse. 

Entering positive definite, Hadamard matrix or covariance matrix, Wolfram Alpha doesn’t find anything, which means that Wolfram Alpha is either in a beta stage or is not very focussed on matrix matters.

In contrast to Google, Wolfram Alpha can compute results and present them with graphs which is a major step forward and places Wolfram Alpha as more of a knowledge gateway than a search engine. But there seem to be quite a gap from the ambitions and the present level of completeness.

Tim Davis of the University of Florida is maintaining an amazing collection of sparse matrices with applications in many diverse scientific fields such as geometry, fluid dynamics, acoustics, networks, finance and what not. And these sparse matrices have beautiful visual representations.  

Thanks to many good contributions, we are now able to provide you with an updated and improved version of The Matrix Cookbook. You can download it from the link to the right as usual …

PS: We have decided to turn off direct comments on the blog because of massive problems with spam - sorry. Please send your comments to “cookbook (at) 2302 (dot) dk”. Thanks!

Today it is the four years ago The Matrix Cookbook started. Since then it has been used, cited and downloaded by numerous people from very different fields such as statistics, machine learning, quantum mechanics, chemistry and many engeneering disciplines.We thank everybody for the interest in The Matrix Cookbook and wish that you, the readers, will continue to help us develop it and keep it alive.

The Matrix Cookbook has been updated with correction of typos and some minor additions on

• Idempotent matrices
• Hermitian, and skew-hermitian
• Stochastic, aka Transition, aka Probability matrices

We sincerely hope you will benefit from the cookbook and join the community effort by reporting typos and submit suggestions.

Thanks to Professor Ralf Östermark, Åbo Akademi University, we have become aware of an error in the derivative of Mixtures of Gaussian in section 8.4.2 (version sep 2007). The derivative of the density with respect to the mean vector, should not have a minus sign, and the correct formula are:

 It is equation (315) in the image above which have been corrected (above formula is correct).

We apologize for the inconvenience and will of course make sure the error will be corrected in the next updated version of “The Matrix Cookbook”.

As we were gearing up to celebration of download number 100.000, things suddenly changed fast: In one hectic week several hundred thousands downloads erupted out of nowhere, boosting the total far beyond 400.000. Looking errornous or at least unregular in some way, we are looking into what happened, but have not yet finished that part.

We are happy to announce that The Matrix Cookbook has been updated for a September 2007 version. This version includes new material on topics like the Schur complement, Cramers rule, othorgonal matrices, matrix- and vector norms, etc. Also a few errors has been corrected.  

The Matrix Cookbook has now passed to 70K mark. That is, since the announcement of the first version in 2004, The Matrix Cookbook has been downloaded by users all over the world more than 70,000 times. That makes the document the most downloaded document in history, at the institute “Informatics and Mathematical Modelling”, Technical University of Denmark.

Al though, in all fairness, it should be said that some of these hits are the same people downloading it more than once, it is true that The Matrix Cookbook has become a success in many branches of science from quantum mechanics, to econometrics and neurocomputations. We, the proud authors, are wishing all users the best of luck with the matrix computations around the world.

We have received a question from Nathan Litke, which you all are most welcome to help out on. It is regarding so-called adjugate matrices:  

“I need the derivative of the adjugate matrix, e.g. Adj([a b; c d]) = [d -b; -c a].  Are you aware of a formula for this derivative, which is valid for non-singular and singular matrices?”