The Gram-Schmidt process replaces linearly independent vectors by orthogonal vectors with the same successive spans, subtracting from each its projections onto the preceding vectors.
A reduced QR decomposition of a full-column-rank matrix writes , where has orthonormal columns and is upper triangular with positive diagonal.
For a full-column-rank matrix , extend the thin QR decomposition to an orthogonal matrix . Thenso maps the column space of onto the first coordinate directions. Successive Householder transformations construct the same reduction without first forming a full orthogonal basis.
A linear least-squares problem minimizes over . If has full column rank, orthogonality reduces it to the upper-triangular system formed from the leading block of and the leading entries of .
For a symmetric bilinear form , the same projection formulaproduces a -orthogonal basis whenever the chosen pivots are nonzero. Reordering the input vectors can avoid a zero pivot when a suitable anisotropic vector remains.
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The Gram–Schmidt process is an algorithm used in linear algebra to orthogonalize a set of vectors in an inner product space, most commonly in Euclidean space. The primary goal of this process is to take a finite, linearly independent set of vectors and transform it into an orthogonal (or orthonormal) set of vectors, which are mutually perpendicular to one another or normalized to have unit length.