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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.
More generally, we can factor a complex m×n matrix A, with m ≥ n, as the product of an m×m unitary matrix Q and an m×n upper triangular matrix R.As the bottom (m−n) rows of an m×n upper triangular matrix consist entirely of zeroes, it is often useful to partition R, or both R and Q:
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.
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.
The projection pursuit method uses Gram-Schmidt orthogonalization to ensure the independence of the extracted signal, while ICA use infomax and maximum likelihood estimate to ensure the independence of the extracted signal. The Non-Normality of the extracted signal is achieved by assigning an appropriate model, or prior, for the signal.
Linear multistep method — the other main class of methods for initial-value problems Backward differentiation formula — implicit methods of order 2 to 6; especially suitable for stiff equations Numerov's method — fourth-order method for equations of the form y ″ = f ( t , y ) {\displaystyle y''=f(t,y)}
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 ...