抽象的

More on the orthogonal complement functions

Robert Jennrich and Albert Satorr

Continuous orthogonal complement functions have had an interesting history in covariance structure analysis. They were used in a seminal paper by Browne in his development of a distribution-free goodness of fit test for an arbitrary covariance structure. The proof of his main result Proposition 4 used a locally continuous orthogonal complement function, but because he failed to show such functions existed his proof was incomplete. In spite of the fact that his test had been used extensively, this problem was not noticed until 2013 when Jennrich and Satorra pointed out that his proof was incomplete and completed it by showing that locally continuous orthogonal complement functions exist. This was done using the implicit function theorem. A problem with the implicit function approach is that it does not give a formula for the locally continuous function produced. This problem was potentially solved by Browne and Shapiro who gave a very simple formula F(X) for an orthogonal complement of X. Unfortunately, they failed to prove that their function actually produced orthogonal complements. We will prove that given a p×q matrix X0 with full column rank q