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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.
The Gram–Schmidt procedure gives a basis of orthogonal polynomials in which the operator: ¯ has a tridiagonal Jacobi matrix representation. This in turn leads to a tridiagonal model of positive Hankel kernels.
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.
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.
An alternative way to arrive at the same expressions is to take the first three derivatives of the curve r′(t), r′′(t), r′′′(t), and to apply the Gram-Schmidt process. The resulting ordered orthonormal basis is precisely the TNB frame. This procedure also generalizes to produce Frenet frames in higher dimensions.
In linear algebra, orthogonalization is the process of finding a set of orthogonal vectors that span a particular subspace.Formally, starting with a linearly independent set of vectors {v 1, ... , v k} in an inner product space (most commonly the Euclidean space R n), orthogonalization results in a set of orthogonal vectors {u 1, ... , u k} that generate the same subspace as the vectors v 1 ...