Vector transpose

The transpose operator indicates an interchange of rows with columns and vice versa. If m is a (column) vector, then a row vector is indicated by u.  

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The null space KerB is a vector subspace of the whole space of Af-vectors. Let us have M-L linearly independent vectors (transposed row vectors) oej e KerB (A = 1, —, M-L), thus a basis of KerB . The basis can be completed by some L linearly independent vectors, say Bk (A = 1, —, L) to a basis of. Then the matrix... 

The nature of the function / is the next step in problem classification. Many application areas such as finance and management-planning tackle linear or quadratic objective functions. These can be written in vector form, respectively, as /(x) = b x -I- fa and /(x) = x Ax -i- b x + fa where b is a column vector, fa is a scalar, and A is a constant symmetric x matrix (i.e., one whose entries satisfy A,j = Ay, ). The superscripts above refer to a vector transpose thus x y is an inner product. Linear programming problems refer to linear objective functions subject to linear constraints (i.e., a system of linear equations), and quadratic programming problems have quadratic objective functions and linear constraints. 

The square brackets denote a vector, and [ ] a transposed vector. The exact expression for the Onsager-Machlup action is now approximated by... 

Here tp denotes the conjugate transpose of ip. Another conserved quantity is the norm of the vector ip, i.e., ip ip = const, due to the unitary propagation of the quantum part. 

As indicated earlier, the vaUdity of the method of dimensional analysis is based on the premise that any equation that correcdy describes a physical phenomenon must be dimensionally homogeneous. An equation is said to be dimensionally homogeneous if each term has the same exponents of dimensions. Such an equation is of course independent of the systems of units employed provided the units are compatible with the dimensional system of the equation. It is convenient to represent the exponents of dimensions of a variable by a column vector called dimensional vector represented by the column corresponding to the variable in the dimensional matrix. In equation 3, the dimensional vector of force F is [1,1, —2] where the prime denotes the matrix transpose. 

Theorem 5. The transpose of is a complete B-matrrx of equation 13. It is advantageous if the dependent variables or the variables that can be regulated each occur in only one dimensionless product, so that a functional relationship among these dimensionless products may be most easily determined (8). For example, if a velocity is easily varied experimentally, then the velocity should occur in only one of the independent dimensionless variables (products). In other words, it is sometimes desirable to have certain specified variables, each of which occurs in one and only one of the B-vectors. The following theorem gives a necessary and sufficient condition for the existence of such a complete B-matrix. This result can be used to enumerate such a B-matrix without the necessity of exhausting all possibilities by linear combinations. 

In accordance with Section 9.1, we represent a vector z as an ordered vertical arrangement of numbers. The transpose then represents an ordered horizontal arrangement of the same numbers. The dimension of a vector is equal to the number of its elements, and a vector with dimension n will be referred to as an -vector. 

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... 

PAHs introduced in Section 34.1. A PCA applied on the transpose of this data matrix yields abstract chromatograms which are not the pure elution profiles. These PCs are not simple as they show several minima and/or maxima coinciding with the positions of the pure elution profiles (see Fig. 34.6). By a varimax rotation it is possible to transform these PCs into vectors with a larger simplicity (grouped variables and other variables near to zero). When the chromatographic resolution is fairly good, these simple vectors coincide with the pure factors, here the elution profiles of the species in the mixture (see Fig. 34.9). Several variants of the varimax rotation, which differ in the way the rotated vectors are normalized, have been reviewed by Forina et al. [2]. 

Transposing both sides to column vectors gives a set of q linear equations in p unknowns... 

The functions tpi(x) are, in general, complex functions. As a consequence, ket space is a complex vector space, making it mathematically necessary to introduce a corresponding set of vectors which are the adjoints of the ket vectors. The adjoint (sometimes also called the complex conjugate transpose) of a complex vector is the generalization of the complex conjugate of a complex number. In Dirac notation these adjoint vectors are called bra vectors or bras and are denoted by or (/. Thus, the bra (0,j is the adjoint of the ket, ) and, conversely, the ket j, ) is the adjoint (0,j of the bra (0,j... 

MATLAB returns the results in a column vector. Most functions in MATLAB take either row or column vectors and we usually do not have to worry about transposing them. 

The target gene is markedwith yellow, transposon is blue and the shuttle vector is indicated with a circle. The vector delivering the transposon will be eliminated after the random insertion of the transposable element. Insertion of the transposon inactivates the target gene function this can be corrected via introduction of the gene on a vector (complementation). 

The transpose of a matrix or a vector is formed by assembling the elements of the first row of the matrix as the elements of the first column of the transposed matrix, the second row into the second column, and so on. In other words, atj in the original matrix A becomes the component aji in the transpose Ar. Note that the position of the diagonal components atj) are unchanged by transposition. If the dimension of A is n X m, the dimension of Ar is m X n (m rows and n columns). If square matrices A and AT are identical, A is called a symmetric matrix. The transpose of a vector x is a row... 

R.3 in (3.1) is the transpose of the third column of R, and represents a unit vector directed along the static field B0. More generally R j, i = 1,2,3, denotes the ith column of R. This notation proved to be convenient in the description of ENDOR spectra, especially in more complex cases, e.g. in CP ENDOR, PM-ENDOR, DOUBLE ENDOR as well as in spin decoupling experiments (Sect. 4). 

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