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The Matrix Cookbook - DTU

The Matrix Cookbook [ ]Kaare Brandt PetersenMichael Syskind PedersenVersion: November 15, 20121 IntroductionWhat is this?These pages are a collection of facts (identities, approxima-tions, inequalities, relations, ..) about matrices and matters relating to is collected in this form for the convenience of anyone who wants a quickdesktop reference .Disclaimer:The identities, approximations and relations presented here wereobviously not invented but collected, borrowed and copied from a large amountof sources. These sources include similar but shorter notes found on the internetand appendices in books - see the references for a full :Very likely there are errors, typos, and mistakes for which we apolo-gize and would be grateful to receive corrections at ongoing:The project of keeping a large repository of relations involvingmatrices is naturally ongoing and the version will be apparent from the date inthe :Your suggestio

Keywords: Matrix algebra, matrix relations, matrix identities, derivative of determinant, derivative of inverse matrix, di erentiate a matrix. Acknowledgements: We would like to thank the following for contributions and suggestions: Bill Baxter, Brian …

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The Matrix Cookbook

www2.imm.dtu.dk

A B Hadamard (elementwise) product A B Kronecker product 0 The null matrix. Zero in all entries. I The identity matrix Jij The single-entry matrix, 1 at (i;j) and zero elsewhere A positive de nite matrix A diagonal matrix Petersen & Pedersen, The Matrix Cookbook, Version: November 15, 2012, Page 5

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72.DESCENT METHODS Definition 2.6. Descent direction. h is a descent direction for Fat x if h>F0(x) <0: If no such hexists, then F0(x)=0, showing that in this case xis stationary. Otherwise, we have to choose fi, ie how far we should go from x in the direction given by hd, so that we get a decrease in the value of the objective function.

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properties and a natural connection to clustering. Some potential appli-cations and illustrative examples are discussed. 1 Introduction In many areas of artificial intelligence, information retrieval, and data min-ing, one is often confronted with intrinsically low-dimensional data lying in a very high-dimensional space.

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