N-vector

Veitstanz, m. St. Vitus s dance, chorea. Vektorengleichung,/. vector equation. Vektorgeriist, n. vector diagram, vektorieil, a. vectorial. 

A set of n vectors of dimension n which are linearly independent is called a basis of an -dimensional vector space. There can be several bases of the same vector... 

Every n vector can be represented as a point in an -dimensional coordinate space. The n elements of the vector are the coordinates along n basis vectors, such as defined in the previous section. The null vector 0 defines the origin of the coordinate space. Note that the origin together with an endpoint define a directed line segment or axis, which also represents a vector. Hence, there is an equivalence between points and axes, which can both be thought as geometrical representations of vectors in coordinate space. (The concepts discussed here are extensions of those covered previously in Sections 9.2.4 to 9.2.5.)... 

On the other hand, one can also regard the same matrix X as an ordered array of n vectors of dimension p, each of the form ... 

Fig. 29.5. Geometrical interpretation of an nxp matrix X as either a row-pattern of n points P" in p-dimensional column-space S (left panel) or as a column-pattern of p points / in n-dimensional row-space S" (right panel). The p vectors Uy form a basis of 5 and the n vectors v, form a basis of 5".

In the matrix-by-vector product one postmultiplies an nxp matrix X with a p vector y which results in an n vector z ... 

The vector-by-matrix product involves an n vector which premultiplies an nxp matrix Y to yield a p vector z ... 

The outer product results from premultiplying an n vector x with a p vector... 

The matrix-to-vector product can be interpreted geometrically as a projection of a pattern of points upon an axis. As we have seen in Section 29.4 on matrix products, if X is an nxp matrix and if v is a p vector then the product of X with v produces the n vector s ... 

In multiple linear regression (MLR) we are given an nxp matrix X and an n vector y. The problem is to find an unknown p vector b such that the product y of X with b is as close as possible to the original y using a least squares criterion ... 

In NIPALS one starts with an initial vector t with n arbitrarily chosen values (Fig. 31.12). In a first step, the matrix product of the transpose of the nxp table X with the n-vector t is formed, producing the p elements of vector w. Note that in the traditional NIPALS notation, w has a different meaning than that of a weighting vector which has been used in Section 31.3.6. In a second step, the elements of the p-vector w are normalized to unit sum of squares This prevents values from becoming too small or too large for the purpose of numerical computation. The... 

The r coordinates of the variable point which traces out the trajectory of the yth column-item in the r-dimensional biplot are compiled in the r-vector Sj. The elements of the latter can be estimated by means of linear regression of the nxr factor scores S upon an n-vector which is defined as ... 

Another use for the matrix inverse is to express one set of variables in terms of another, an important operation in constrained optimization (see Chapter 8). For example, suppose x and z are two n- vectors that are related by... 

AnnXn matrix has n eigenvalues. We define an n-vector v, the eigenvector, which is associated with an eigenvalue e such that... 

In contrast, Ni[pz(iV-Me2)8] C60 (lOOd) has a totally different supramolecular structure featuring a noncentrosymmetric 1 1 complex with a strongly warped pz unit. The plane of diametrically opposite pyrrole rings is being rotated by -30° with respect to each other about their N—N vectors with the nickel and the four coordinated nitrogen atoms are coplanar to within 0.006 A. Each C60 unit lies within a cleft formed by two steeply inclined (by -12°) Ni-pz molecules (Fig. 33), the shortest Ni—C (C60) distances are 3.29 and 3.64 A. 

The MD trajectories of 1018 ps were computed at T = 303 K. The zero point (z = 0) is the centre of the bilayer it was calculated during the MD simulations as the middle point between the centres of P-N vectors of the two monolayers (the centre of the P-N vector is the middle of P-N distance). The ratio between the CPU times (MD/SCF) needed to obtain these results is in the order of 105... 

A base of 91" is a set of n vectors such that any vector in 91" can be represented by a linear combination of the base vectors. 

We now return to the very basic concept of vector length in vector spaces. In a Euclidian space spanned by a base of n orthogonal unit vectors eb the squared length l2 of a n-vector t> is the quadratic form given by... 

The idea of a vector space is usefully extended to an infinite number of dimensions for continuous functions. Given a function /(e.g.,/ = sinx) and a definition domain (e.g., 0 to In), the coordinates of / = sin x will be the infinite number of values of the function over the definition domain. This definition is consistent with that of Euclidian spaces if a metric is defined. In about the same way as the squared norm of the n-vector x(xux2,. .., x ) is... 

Let u(t) be an n-vector with components (u1,u2,..u ) depending on a single scalar variable t. The derivative of the vector u with respect to the scalar t is the n-vector defined as... 

The gradient vector grad f(x) of the scalar function f(x) is an n-vector defined as... 

X is an n-vector of n independent normal standardized variables, i.e., with zero mean, unit variances and null covariances. Find (i) the density function and (ii) the covariance matrix of the vector Y given by... 

A random n-vector X has a mean vector fi and an n x n covariance matrix . a is the diagonal matrix with standard deviations as diagonal terms and p the correlation matrix. Find the correlation matrix of the reduced vector given by... 

If we now consider an n-vector X of n random variables ( data ) with mean px and covariance matrix related to a vector Y of m ancillary variables through i= 1,..., m functions q>t... 

A system of - -1 particles of masses Mi and charges g,- may be described at any point in time by the n+ vectors, describing the positions of the particles ... 

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