Transpose matrices

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. 

A data matrix with column-wise organization is easily converted to row-wise organization by taking its matrix transpose, and vice versa. If you are not familiar with the matrix transpose operation, please refer to the discussion in Appendix A. 

To (hopefully) help keep things simple, we will organize all of our data into column-wise matrices. Later on, when we explore Partial Least-Squares (PLS), we will have to remember that the PLS convention expects data to be organized row-wise. This isn t a great problem since one convention is merely the matrix transpose of the other. Nonetheless, it is one more thing we have to remember. 

The transpose of a a matrix is formed by changing each column into a row (or each row into a column). The matrix transpose of a matrix, A is denoted by the superscript T to give AT. 

Integrating the linearized equations along the fiducial trajectory yields the tangent map M(zo,t) which takes the initial variables Zm into the time-evolved variables Z(t) = MZin. Let A be a matrix A = lim (MM)1/2, where M denotes the matrix transpose of M. The... 

FIGURE A.2.1 Matrix transpose and special types of matrices. 

Due to the special structure of MATLAB, readers should be familiar with the mathematical concepts pertaining to matrices, such as systems of linear equations, Gaussian elimination, size and rank of a matrix, matrix eigenvalues, basis change in n-dimensional space, matrix transpose, etc. For those who need a refresher on these topics there is a concise Appendix on linear algebra and matrices at the end of the book. 

One must remember that the product AB is defined only if the number of columns of matrix A is equal to the number of rows of matrix B. If the rowp and columns in matrix A interchange their places, it produces matrix A, which is called a matrix transposed with respect to matrix A... 

In this section, lowercase boldface type denotes a column vector and uppercase boldface type a matrix, and the superscript T denotes matrix transpose.) Multiple mixture spectra with varying analyte concentrations can be written together in matrix form as... 

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... 

Equation (9.24) can be expanded by recalling that the matrix transpose process is additive [2]... 

We start by developing the quadratic terms in Eq. (A2.80), for a given atom. The vector components along the Cartesian coordinate scheme used to generate the root mean square displacement vectors are developed (a superscripted T denotes matrix transpose). 

T superscript in matrix, transposed TCC tricarboxylic acid cycle pR reaction volume (L)... 

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