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The first two steps of the Gram–Schmidt process. In mathematics, particularly linear algebra and numerical analysis, the Gram–Schmidt process or Gram-Schmidt algorithm is a way of finding a set of two or more vectors that are perpendicular to each other.
A practical way to enforce this is by requiring that the next search direction be built out of the current residual and all previous search directions. The conjugation constraint is an orthonormal-type constraint and hence the algorithm can be viewed as an example of Gram-Schmidt orthonormalization. This gives the following expression:
This resembles the problem of orthogonalization, which requires = for any , and = for any =. Thus the problem of finding conjugate axes is less constrained than the problem of orthogonalization, so the Gram–Schmidt process works, with additional degrees of freedom that we can later use to pick the ones that would simplify the computation:
This method has greater numerical stability than the Gram–Schmidt method above. The following table gives the number of operations in the k -th step of the QR-decomposition by the Householder transformation, assuming a square matrix with size n .
The Gram-Schmidt theorem, together with the axiom of choice, guarantees that every vector space admits an orthonormal basis. This is possibly the most significant use of orthonormality, as this fact permits operators on inner-product spaces to be discussed in terms of their action on the space's orthonormal basis vectors.
A Gram–Schmidt process could orthogonalize the columns, but it is not the most reliable, nor the most efficient, nor the most invariant method. The polar decomposition factors a matrix into a pair, one of which is the unique closest orthogonal matrix to the given matrix, or one of the closest if the given matrix is singular.
In numerical linear algebra, the Arnoldi iteration is an eigenvalue algorithm and an important example of an iterative method.Arnoldi finds an approximation to the eigenvalues and eigenvectors of general (possibly non-Hermitian) matrices by constructing an orthonormal basis of the Krylov subspace, which makes it particularly useful when dealing with large sparse matrices.
Applying Gram–Schmidt one obtains an orthonormal basis (e i) for H. Let (H i) be the corresponding nested sequence of "coordinate" subspaces of H. The matrix a i,j expressing T with respect to (e i) is almost upper triangular, in the sense that the coefficients a i+1,i are the only nonzero sub-diagonal coefficients.