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Gram-Schmidt Process

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Gram-Schmidt process is a way to convert a set of non-orthogonal vectors into a set of orthogonal vectors, while preserving the subspaces spanned by initial vectors. Gram-Schmidt process in Euclidean spaces is captured in a computational algorithm called QR-decomposition.

QR-decomposition is used in many computational algorithms as an intermediate step, most notably, in calculating eigenvalues of matrices (the QR-algorithm).

In the context of statistics, Gram-Schmidt process and QR-algorithm have many applications:

  1. A proof that Studentized means have Student t-distribution.
  2. Solution of least squares problems.
  3. Construction of orthogonal contraststs, such as orthogonal polynomial contrasts.

The attached documents contain an exposition of Gram-Schmidt process, QR-decomposition and a statistical application to Studentized means.