v2007.09.17 - Convex Optimization

Appendix BSimple matricesMathematicians also attempted to develop algebra of vectors butthere was no natural definition of the product of two vectorsthat held in arbitrary dimensions. The first vector algebra thatinvolved a noncommutative vector product (that is, v×w need notequal w×v) was proposed by Hermann Grassmann in his bookAusdehnungslehre (1844). Grassmann’s text also introduced theproduct of a column matrix and a row matrix, which resulted inwhat is now called a simple or a rank-one matrix. In the late19th century the American mathematical physicist Willard Gibbspublished his famous treatise on vector analysis. In that treatiseGibbs represented general matrices, which he called dyadics, assums of simple matrices, which Gibbs called dyads. Later thephysicist P. A. M. Dirac introduced the term “bra-ket” for whatwe now call the scalar product of a “bra” (row) vector times a“ket” (column) vector and the term “ket-bra” for the product of aket times a bra, resulting in what we now call a simple matrix, asabove. Our convention of identifying column matrices and vectorswas introduced by physicists in the 20th century.−Marie A. Vitulli [277]2001 Jon Dattorro. CO&EDG version 2007.09.17. All rights reserved.Citation: Jon Dattorro, Convex Optimization & Euclidean Distance Geometry,Meboo Publishing USA, 2005.519