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The calculation of the sequence , …, is known as Gram–Schmidt orthogonalization, and the calculation of the sequence , …, is known as Gram–Schmidt orthonormalization. To check that these formulas yield an orthogonal sequence, first compute u 1 , u 2 {\displaystyle \langle \mathbf {u} _{1},\mathbf {u} _{2}\rangle } by substituting the ...
In linear algebra, the Schmidt decomposition (named after its originator Erhard Schmidt) refers to a particular way of expressing a vector in the tensor product of two inner product spaces. It has numerous applications in quantum information theory , for example in entanglement characterization and in state purification , and plasticity .
In other words, the sequence is obtained from the sequence of monomials 1, x, x 2, … by the Gram–Schmidt process with respect to this inner product. Usually the sequence is required to be orthonormal , namely, P n , P n = 1 , {\displaystyle \langle P_{n},P_{n}\rangle =1,} however, other normalisations are sometimes used.
The Hilbert–Schmidt operators form a two-sided *-ideal in the Banach algebra of bounded operators on H. They also form a Hilbert space, denoted by B HS (H) or B 2 (H), which can be shown to be naturally isometrically isomorphic to the tensor product of Hilbert spaces, where H ∗ is the dual space of H.
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. What results is a deep ...
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.
The concept of orthogonality may be extended to a vector space over any field of characteristic not 2 equipped with a quadratic form .Starting from the observation that, when the characteristic of the underlying field is not 2, the associated symmetric bilinear form , = ((+) ()) allows vectors and to be defined as being orthogonal with respect to when (+) () = .
The SVD also plays a crucial role in the field of quantum information, in a form often referred to as the Schmidt decomposition. Through it, states of two quantum systems are naturally decomposed, providing a necessary and sufficient condition for them to be entangled : if the rank of the Σ {\displaystyle \mathbf {\Sigma } } matrix is larger ...