Vector inner product

The vector inner product is the same as the familiar dot product of physical vectors. The following are equivalent notations, where jc is an n-element vector and y is an n-element veetor  

The inner product is always the result of multiplying a row vector by a column vector (the reverse is called the outer product and is discussed later). The result of the inner product is a scalar (a single number). 

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This section sets the stage for some of the ideas discussed later. Going from equations with discrete vectors and matrices to equations with functions in its basic form is not difficult. Let us demonstrate these ideas by looking at an example where PCA is applied to functions. Let X be a matrix of continuous spectra. This means that the N rows in X are really functions such that X = [xi(t) X2(t) ... xisi(t) ]. One way to find the principal components of X is to solve the eigenquation of the covariance matrix G = XX. For the discrete case G can be written in terms of vector inner products... 

Using the distributive properties of inner and outer products, any products of linear combinations can be expanded into linear combinations of products. Thus any scalar-valued expression can be expanded into a polynomial in the inner products of pairs of vectors, bivectors, or trivectors, together with scalars and triple products. Inner products of bivectors and trivectors can be further expanded into polynomials in the inner products of vectors only, using the equivalence to Grami-ans derived in the previous section. Moreover, if two triple products occur in any term, we can likewise expand them as a Gramian into a polynomial in the vector inner products, since... 

The vector inner product has an interesting and useful geometric interpretation. The angle 6 between two nonzero vectors u and v is given by the following (which is easily derived from the law of cosines for the two-dimensional case) ... 

According to the Helmholtz theorem the Hilbert space of 2-D vector fields p x, y) with the inner product... 

This construction requires one matrix-vector multiplication with S and two inner products in each recursive step. Therefore, it is not necessary to store S explicitly as a matrix. The Lanczos process yields the approximation [21, 7, 12]... 

Linear operators in finite-dimensional spaces. It is supposed that an n-dimensional vector space R is equipped with an inner product (, ) and associated norm a = / x x). By the definition of finite-dimensional space, any vector x G i n can uniquely be represented as a linear combina-... 

The inner product in the space Q /j of all grid vector-functions given on the grid ijjp and vanishing on the boundary is defined by... 

In this and subsequent sections we will make frequent use of the scalar product (also called inner product) between two vectors x and y with the same dimension n, which is defined by ... 

The projection of the X data vectors onto the first eigenvector produces the first latent variable or pseudomeasurement set, Zx. Of all possible directions, this eigenvector explains the greatest amount of variation in X. The second eigenvector explains the largest amount of variability after removal of the first effect, and so forth. The pseudomeasurements are called the scores, Z, and are computed as the inner products of the true measurements with the matrix of loadings, a ... 

The scalar (or inner) product of two vectors is defined by the relation... 

A vector space L defined over a field F is further called an inner-product space or unitary space if its elements satisfy one more condition ... 

The ordinary three-dimensional space of position vectors is also an inner product space with the familiar rule for taking the scalar product of two vectors. 

The distance between two vectors (or points, or functions) x and y defined in terms of an inner product is... 

Equation (A.7) is referred to as the inner product, or dot product, of two vectors. If the two vectors are orthogonal, then xTy = 0. In two or three dimensions, this means that the vectors x and y are perpendicular to each other. 

As in the case of regular vectors, the components a and b can be found by forming the inner products upon multiplication of the /(x) expansion by the adequate sine or cosine... 

PVVaik) is therefore the direction of constrained minimization. As in the case of Lagrange multipliers, no progress can be made and search will stop when the (k + l)th minimization direction PV (k+1) is orthogonal to the fcth minimization direction PV inner product of these vectors becomes less than an arbitrarily small value. 

The scalar product (dot product, inner product) requires two vectors of the same length the result is a scalar obtained by pair-wise multiplication of... 

Directional or Lie Derivative. This operation is defined as the inner product between the partial derivative of a smooth real-valued function and a smooth vector field. As a result, a new smooth real-valued function is obtained. For example, the directional derivative of x) along the vector field f x) is given by... 

Lie Product or Braked. The second operation involves two smooth vector fields e.g., f x) and g x) both defined on an open set U o/M". From this operation, a new smooth vector field is constructed by the following inner product... 

Feasible x) and y) give upper and lower bounds on the optimal value of the objective function, which in the 2-RDM problem is the ground-state energy in a finite basis set. The primal and dual solutions, x) and y), sie feasible if they satisfy the primal and dual constraints in Eqs. (107) and (108), respectively. The difference between the feasible primal and the dual objective values, called the duality gap fi, which equals the inner product of the vectors x) and z). 

Note the similarity of Eqs. 2.43 and 2.44 with Eqs. 2.80 and 2.81 because both the vectors in the former equations and the functions of the latter are all elements of linear vector spaces. The main difference arises in the way in which the inner products are evaluated. Also, as was the case for vectors, if the field functions are non-negative functions, SCar(F, F pj will be non-negative. When this is not the case, however, Sr.ir(F (,F g) may become negative, a situation that also obtains for the other similarity indices discussed in the remainder of this section. Maggiora et al. (43) have treated this case in great detail for continuous field functions, but the arguments can be carried through for finite vectors as well (vide supra). 

In Subheading 2.3. the important class of vectors with continuous-valued components is described. A number of issues arise in this case. Importantly, since the objects of concern here are vectors, the mathematical operations employed are those applied to vectors such as addition, multiplication by a scalar, and formation of inner products. While distances between vectors are used in similarity studies, inner products are the most prevalent type of terms found in MSA. Such similarities, usually associated with the names Carbo and Hodgkin, are computed as ratios, where the inner product term in the numerator is normalized by a term in the denominator that is some form of mean (e.g., geometric or arithmetic) of the norms of the two vectors. 

Problem 5-4. (a) Express the norm of a vector in terms of an inner product, (b) Express the cosine of the angle between two vectors in terms of inner products. 

Problem 5-5. A point on the a —axis is represented by a vector of the form X, 0,0), where X is a real number, (a) Eind the general form of the vectors that represent points of the 2/— and 2 —axes, (b) Show that the inner product of any vector representing a point on the a —axis with any vector representing a point on the 2/— or z—axes is zero. 

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