Quick Constructions of Non-Trivial Real Symmetric Idempotent Matrices

In the theory and applications of linear models, idempotent matrices appear frequently. It remains important to construct non-trivial quadratic forms distributed as chi-squares. However, quick and easy methods to explicitly construct such matrices with all non-zero entries are hard to come by. We include two direct methods to explicitly exhibit nxn real symmetric idempotent matrices of rank r with all non-zero entries. Then, we exhibit explicit non-trivial real symmetric idempotent matrices A; B such that AB = O:

DOI: http://dx.doi.org/10.4038/sljastats.v15i1.6794