Vector space orthonormal basis

In any Hilbert space the basis vectors can always be chosen to be orthonormal ... 

Transformations in Hilbert Space.—Consider any vector /> in with components with respect to some orthonormal basis... 

A set of complete orthonormal functions ipfx) of a single variable x may be regarded as the basis vectors of a linear vector space of either finite or infinite dimensions, depending on whether the complete set contains a finite or infinite number of members. The situation is analogous to three-dimensional cartesian space formed by three orthogonal unit vectors. In quantum mechanics we usually (see Section 7.2 for an exception) encounter complete sets with an infinite number of members and, therefore, are usually concerned with linear vector spaces of infinite dimensionality. Such a linear vector space is called a Hilbert space. The functions ffx) used as the basis vectors may constitute a discrete set or a continuous set. While a vector space composed of a discrete set of basis vectors is easier to visualize (even if the space is of infinite dimensionality) than one composed of a continuous set, there is no mathematical reason to exclude continuous basis vectors from the concept of Hilbert space. In Dirac notation, the basis vectors in Hilbert space are called ket vectors or just kets and are represented by the symbol tpi) or sometimes simply by /). These ket vectors determine a ket space. 

It should be clear that the set of all real orthogonal matrices of order n with determinants +1 constitutes a group. This group is denoted by 0(n) and is a continuous, connected, compact, n(n — l)/2 parameter9 Lie group. It can be thought of as the set of all proper rotations in a real n-dimensional vector space. If xux2,. ..,xn are the orthonormal basis vectors in this space, a transformation of 0(n) leaves the quadratic form =1 x invariant. 

The occupation number vectors are thus the common eigenvectors for the hermitian and commuting set of operators (aj a agag,. -.a am) and there is a one to one correspondence between an occupation vector and a set of eigenvalues for (aj av a ag,- -.a am). This is consistent with the definition of the occupation number vectors as being an orthonormal basis for the Fock space. 

We start by constructing an orthonormal basis a, b, c (where c is a unit vector along the wavevector k, with a and b in the two-dimensional vector space orthogonal to k). The significance of this ansatz is that any vector function F(x,y,z) is divergence free if and only if its Fourier coefficients F(k) are orthogonal to k, that is if k -F(k) =0. Thus, F(k) is a linear combination of a(k) and b(k). Lesieur defines the complex helical waves as... 

The special orthogonal group SO(2) is the group of proper rotations in the 2-D space of real vectors, ft2, about an axis z normal to the plane containing x and y. Since there is only one rotation axis z, the notation Rif z) for the rotation of the unit circle in ft2 will be contracted to Rif). Then, for the orthonormal basis (ei e2, ... 

This algorithm, standard in the widely used GAUSSIAN program system, is a rank-m update of the Hessian matrix, in an orthonormal basis [356], A basis of unit vectors is constructed in the m -dimensional vector space spanned by the increments Aq. For k = 1, m, define... 

The symbols a and p are the ones most familiar to chemists. For the definition of spin operators, it is convenient to utilize the representation of the spin eigenfunctions as the orthonormal basis vectors of a two-dimensional (2D) vector space. In this representation, the spin operators may be written as matrices... 

The eigenvectors in Vt can be used to form a set of orthonormal row basis vectors for A. The eigenvectors are called loadings or sometimes abstract factors or eigenspectra, indicating that while the vectors form a basis set for the row space of A, physical interpretation of the vectors is not always possible (see Figure 4.2). 

When the true intrinsic rank of a data matrix (the number of factors) is properly determined, the corresponding eigenvectors form an orthonormal set of basis vectors that span the space of the original data set. The coordinates of a vector a in an m-dimensional space (for example, a 1 x m mixture spectrum measured at m wavelengths) can be expressed in a new coordinate system defined by a set of orthonormal basis vectors (eigenvectors) in the lower-dimensional space. Figure 4.14 illustrates this concept. The projection of a onto the plane defined by the basis vectors x and y is given by a. To find the coordinates of any vector on a normalized basis vector, we simply form the inner product. The new vector a, therefore, has the coordinates a, = aTx and a2 = aTy in the two-dimensional plane defined by x and y. 

The coordinates of a vector x in an m dimensional space, e.g., an m x 1 mixture spectrum measured at m = 700 wavelengths, can be expressed in a new coordinate system defined by a set of orthonormal basis vectors (eigenvectors) in the lowerdimensional space. Clearly, we cannot imagine a 700-dimensional space. It is... 

It is not always feasible to directly measure the ancilla independently from the information system in other words, it is sometimes impossible to perform a projection onto disentangled subspaces of H of the form 7T/0Span o ) in some cases, as for the example proposed in Sec. 3, one can only project onto entangled subspaces of the total Hilbert space H. In such a case the information initially stored in the vector ipi) = J2i=i r< Iu<) G Hi must be transferred into an entangled state of X and A of the form ip) = i r> H) where the I vectors 0) (i = 1.. .., I) which form an orthonormal basis of the information-carrying subspace C, are generally not factorized as earlier but entangled states. Nevertheless the same method as before can be used in that case to protect information, albeit in a different subspace C. 

The remaining u,, 1 < i < d, are chosen to be orthogonal to u,/, so that they lie in the allowed hyperplane. Indeed, the u, form an orthonormal basis for the composition space. This orthonormal basis is identified by the Gram-Schmidt procedure. First, the original composition basis vectors are defined... 

Quantum mechanics is generally formulated on the Hilbert space of vectors The functions ij/(q) = and (p) =

are the expansion coefficients of ) in an appropriate complete orthonormal basis, that is,... 

Thus the functions cVv 2iri i and c / /2ir 4 i form an orthonormal basis in the space 2l- The functions xy, yz, xz, x — j/, y — z, z — X are linearly dependent (see Example 2.15), and the vector space of these six functions is only five-dimensional. Only five functions are required to span the space xy, yz, xz, x — y, 3z — r is one such choice of basis vectors as was shown in Example 2.15. As they stand, these functions are not square integrable, but they appear in quantum theory (d orbitals) with each multiplied by the factor e" . 

F. B.2. A pictorial representation of something that surely cannot be represented. Using poetic license, an orthonormal basis in the Hilbert space looks like a hedgehog of the unit vectors (their number equal to oo), each pair rf them orthogonal. This is in analogy to a 2-D or 3-D basis set, where the hedgehog has two or three orthogonal unit vectors. 

Now consider a function space f) and a linear transformation matrix. A, which recombines the basis functions to yield a transformed basis set, say If). Such a linear transformation of an orthonormal vector space preserves orthonormality if and only if the transformation matrix A is unitary. Assuming that A is unitary, the forward implication is easily proven ... 

The individual unit of classical information is the bit an object that can take either one of two values, say 0 or 1. The corresponding unit of quantum information is the quantum bit or qubit. It describes a state in the simplest possible quantum system [1,2]. The smallest nontrivial Hilbert space is two-dimensional, and we may denote an orthonormal basis for the vector space as 0> and 11 >. A single bit or qubit can represent at most two numbers, but qubits can be put into infinitely many other states by a superposition ... 

The x(t) and y t) are said to be orthogonal if their inner product (x(t), y(t) = 0, and if each vector is also a unit vector, the vector set is called orthonormal. The collection of these vectors can span completely any vector in the space. They are called the orthogonal (orthonormal) basis. The inner product can be roughly understood as correlation of x(t) and 3 ( ) the higher their correlation, the larger the absolute value. If they are uncorrelated, then (x(t), y(t) = 0, x(t) and y t) are orthogonal. 

To establish the rotations of the group let us take the basic lattice vectors ai, 02, as as the basis unit vectors in the space of the lattice vectors a, and write down the matrix D R) of the transformation R in the new basis, in which all the lattice vectors have integer components. If the matrix of the orthogonal transformation R in this basis is denoted by D R), then D (R) = U D R)U, where t/ is a matrix of the transformation from the initial orthonormal basis to the basis ai, as, as. If f is a rotation (or mirror rotation) through an angle (p the traces of the matrices D R) and iy R) are equal ... 

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