The biplot graphic display of matrices with application to principal component analysis

TLDR: In this article, a matrix of rank two can be represented as a biplot, which consists of a vector for each row and a column, chosen so that any element of the matrix is exactly the inner product of the vectors corresponding to its row and to its column.

Abstract: SUMMARY Any matrix of rank two can be displayed as a biplot which consists of a vector for each row and a vector for each column, chosen so that any element of the matrix is exactly the inner product of the vectors corresponding to its row and to its column. If a matrix is of higher rank, one may display it approximately by a biplot of a matrix of rank two which approximates the original matrix. The biplot provides a useful tool of data analysis and allows the visual appraisal of the structure of large data matrices. It is especially revealing in principal component analysis, where the biplot can show inter-unit distances and indicate clustering of units as well as display variances and correlations of the variables. Any matrix may be represented by a vector for each row and another vector for each column, so chosen that the elements of the matrix are the inner products of the vectors representing the corresponding rows and columns. This is conceptually helpful in understanding properties of matrices. When the matrix is of rank 2 or 3, or can be closely approximated by a matrix of such rank, the vectors may be plotted or modelled and the matrix representation inspected physically. This is of obvious practical interest for the analysis of large matrices. Any n x m matrix Y of rank r can be factorized as