Vector norms

The usual (Lj) norm of a veetor is the square root of the sum of squares of elements. For a veetor, v, of n elements. 

A less useful norm, called the Lj norm, is the sum of absolute values of the vector elements. 

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An important tool in the study of matrices is provided by vector norms and matrix norms. A vector norm j... is any real valued function of the elements satisfying the following three conditions ... 

This may be expressed by saying that all vector norms (in finitedimensional space) are topologically equivalent otherwise, that if the sequence of vectors xi is such that... 

A square matrix of order n can be considered a vector in a space of n2 dimensions, and if the matrix function j... defines a vector norm in 2-space, it will be called a generalized matrix norm. Thus a generalized matrix norm satisfies... 

A matrix norm is consistent (or compatible) with a given vector norm provided... 

Associated with every vector norm, there is a special consistent matrix norm defined as follows ... 

Since every matrix norm is consistent with some vector norm, the following important theorem follows immediately ... 

The euclidean norm of a matrix considered as a vector in m2-space is a matrix norm that is consistent with the euclidean vector norm. This is perhaps the matrix norm that occurs most frequently in the literature. But the euclidean norm of I is n112 > 1 when n > 1, hence it is not a sup. In fact,... 

Variance, 269 of a distribution, 120 significance of, 123 of a Poisson distribution, 122 Variational equations of dynamical systems, 344 of singular points, 344 of systems with n variables, 345 Vector norm, 53 Vector operators, 394 Vector relations in particle collisions, 8 Vectors, characteristic, 67 Vertex, degree of, 258 Vertex, isolated, 256 Vidale, M. L., 265 Villars, P.,488 Von Neumann, J., 424 Von Neumann projection operators, 461... 

The correlation between the slopes of the planes is expressed by the cosine of the angle between their normal vectors. The normal vector (norm) of a plane is the line perpendicularly to the tangent plane. Here, this vector for i and j is described as follows ... 

The least squares solution x of an unsolvable linear system Ax = b such as our system is the vector x that minimizes the error Ax — 6 in the euclidean vector norm a defined by x Jx +. .. + x% when the vector x has n real entries x. ... 

A graphic that can be produced to better describe the actual situation is the plot of b against y-y, where symbolizes the Euclidean vector norm [4CM16], As Figure 5.14 discloses, an /.-shaped curve results with the best model occurring at the bend, which reflects a harmonious model with the least amount of compromise in the trade-off between minimization of the residual and regression vector norms. The regression vector norm acts as an indicator of variance for the... 

Personal experience has shown that PLS often provides lower RMSEC values than PCR. The improvement in calibration performance must also manifest itself in predictions for independent samples. Therefore, a thorough evaluation of PCR versus PLS in any calibration application must involve using a large external validation data set with a comparison of RMSEPpcr and RMSEPpls in conjunction with respective regression vector norms or other variance expressions. 

We generally denote scalars by lowercase Greek letters (e.g., P), column vectors by boldface lowercase Roman letters (e.g., x), and matrices by capital italic Roman letters (e.g., H). A superscriptT denotes a vector or matrix transpose. Thus xT is a row vector, xTy is an inner product, and AT is the transpose of the matrix A. Unless stated otherwise, all vectors belong to R , the u-dimen-sional vector space. Components of a vector are typically written as italic letters with subscripts (e.g., xux2,.. . , ). The standard basis vectors in R" are the n vectors ei,e2,. . . , e , where e has the entry 1 in the th component and 0 in all others. Often, the associated vector norm is the standard Euclidean norm, j 2, defined as... 

Each of the above vector norms induces a corresponding norm on matrices, through the definition... 

But what is measured in fact in neutron scattering is the differential scattering cross-section, dafcIQ. (q), which is defined as the number of neutrons scattered per second towards a detector in a certain direction per incident beam flux and solid angle. In the case of a liquid or a glass sample for which the average structure is isotropic, only the vector norms (r = r and q= q ) are relevant. 

In Sec. A-l, an abbreviated treatment of matrix and vector norms is presented, and in Sec. A-2, mathematical theorems used in the text are presented. 

The norm of a matrix is said to be compatible with a given vector norm provided Eq. (A-29) is satisfied for any matrix A and any conformable vector X. The following procedure is employed for the construction of the matrix norm such that it is compatible with a given vector norm. First, observe that the lengths of all vectors in any ri-dimensional vector space span the set of all real numbers, and that the process of normalization of each vector throws the lengths of all... 

First it will be shown that for any real nonsingular matrix A of order n, there exists a particular vector Xx such that jXj = 1 and flAXj = A. The matrix norm so constructed is said to be subordinate to the given vector norm. Then it will be shown that the matrix norm so constructed satisfies the four conditions required of a matrix norm as well as the compatibility relationship. Only the matrix norm subordinate to the euclidean vector norm X , given by Eq. (A-26) is considered. Now consider the vector AX, whose vector norm as given by Eq. (A-26), is... 

Now it will be shown that the Hilbert matrix norm and the euclidean vector norm satisfy the four requirements as well as the compatibility relationship. To show that the first condition is satisfied, let A be any real nonsingular, nonzero matrix (A 0). Then there exists a vector X such that X , = 1 and such that the euclidean norm of the vector AX is a maximum and nonzero. Moreover, the particular vector X that satisfies these conditions is the eigenvector Xt corresponding to the largest eigenvalue That is... 

Since there exists an X having a euclidean norm of unity for which the euclidean vector norm of AX is a maximum, it follows that... 

Now let A be any real nonsingular matrix of the same order as Y. Then from the properties of the vector norm, it follows that... 

Since the vector norm of AX is equal to or less than the vector norm of AX t (the Hilbert norm of A = A, = AXt , = it follows that... 

To verify the third condition, consider first any two square matrices A and B which are real and nonsingular. Now pick a vector X0 such that X0 m = 1 and such that the vector norm (A + B)X0, is a maximum. Then the triangle inequality (condition 3 for vector norms) may be used to show that... 

To show that the condition 4 is satisfied, let A and B be any two real nonsingular matrices of the same order. Let X0 denote a vector with a euclidean norm of unity for which the vector norm ABX0 m takes on its maximum value. Then... 

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