Singular value decomposition of matched matrices

We consider the joint analysis of two matched matrices which have common rows and columns, for example multivariate data observed at two time points or split according to a dichotomous variable. Methods of interest include principal components analysis for interval-scaled data, correspondence analysis for frequency data, log-ratio analysis of compositional data and linear biplots in general, all of which depend on the singular value decomposition. A simple result in matrix algebra shows that by setting up two matched matrices in a particular block format, matrix sum and difference components can be analysed using a single application of the singular value decomposition algorithm. The methodology is applied to data from the International Social Survey Program comparing male and female attitudes on working wives across eight countries. The resulting biplots optimally display the overall cross-cultural differences as well as the male-female differences. The case of more than two matched matrices is also discussed.