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Given a pre-Hilbert space , an orthonormal basis for is an orthonormal set of vectors with the property that every vector in can be written as an infinite linear combination of the vectors in the basis. In this case, the orthonormal basis is sometimes called a Hilbert basis for . Note that an orthonormal basis in this sense is not generally a ...
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.
In linear algebra, an orthogonal matrix, or orthonormal matrix, is a real square matrix whose columns and rows are orthonormal vectors. One way to express this is Q T Q = Q Q T = I , {\displaystyle Q^{\mathrm {T} }Q=QQ^{\mathrm {T} }=I,} where Q T is the transpose of Q and I is the identity matrix .
The solution can then be expressed as ^ = (), where is an matrix containing the first columns of the full orthonormal basis and where is as before. Equivalent to the underdetermined case, back substitution can be used to quickly and accurately find this x ^ {\displaystyle {\hat {\mathbf {x} }}} without explicitly inverting R 1 {\displaystyle R ...
If an orthonormal basis is to be produced, then the algorithm should test for zero vectors in the output and discard them because no multiple of a zero vector can have a length of 1. The number of vectors output by the algorithm will then be the dimension of the space spanned by the original inputs.
Find Q minimizing Tr( (Q − M) T (Q − M) ), subject to Q T Q = I. Though written in matrix terms, the objective function is just a quadratic polynomial. We can minimize it in the usual way, by finding where its derivative is zero. For a 3 × 3 matrix, the orthogonality constraint implies six scalar equalities that the entries of Q must satisfy.
The basis functions are typically found by computing the eigenvectors of the covariance matrix of the data set. A more advanced technique is to form a kernel out of the data, using a fixed kernel . The basis functions from the eigenvectors of the kernel matrix are thus non-linear in the location of the data (see Mercer's theorem and the kernel ...
In finite-dimensional spaces, the matrix representation (with respect to an orthonormal basis) of an orthogonal transformation is an orthogonal matrix. Its rows are mutually orthogonal vectors with unit norm, so that the rows constitute an orthonormal basis of V. The columns of the matrix form another orthonormal basis of V.