M x n matrix

Each element is designated with a double subscript in general, an element is called fly where j is its horizontal position in the ith row of the matrix. A matrix with m rows and n elements in each row is an m x n matrix. A square matrix with n elements in each row is an n x n matrix. 

LV Factorization of a Matrix To eveiy m X n matrix A there exists a permutation matrix P, a lower triangular matrix L with unit diagonal elements, and a.nm X n (upper triangular) echelon matrix U such that PA = LU. The Gauss elimination is in essence an algorithm to determine U, P, and L. The permutation matrix P may be needed since it may be necessaiy in carrying out the Gauss elimination to... 

Matrix An m x n matrix A is an array of elements with m rows and n eolumns, and is written as... 

The product of an arbitrary m X n matrix A with a conformable n X p null matrix is the mX p null matrix... 

QR Factorization of a Matrix If A is an m x n matrix with m > n, there exists inmxji unitary matrix = [qij q2,..., q ] and an m X n right triangular matrix R such that A = QR. The OR factorization is frequently used in the actual computations when the other transformations are unstable. 

A is an m X n matrix whose (/, j) element is the constraint coefficient aij9 and c, b, 1, u are vectors whose components are cjy bjt ljy ujy respectively. If any of the Equations (7.7) were redundant, that is, linear combinations of the others, they could be deleted without changing any solutions of the system. If there is no solution, or if there is only one solution for Equation (7.7), there can be no optimization. Thus the... 

Let A be a given (m x n) matrix with m >n and n linearly independent columns. Then there exists a unique (m x m) Q,... 

Noting A the m x n matrix with elements dtpJdXj and 6 the mxm matrix with elements driJdXj yields the compact expression... 

Obviously, the trivial solution v,=0 (/ = L > n) does not fit our needs and we must search for solutions as a constrained problem in which the solution vector is of constant, yet arbitrary, length. In other words, we become interested in the vector with some criterion of best direction regardless of its magnitude, which we may conveniently take as unity. Let us lump the C/ coefficients into the m x n matrix A and the n coefficients Vj into the vector x , hence... 

Let us define X as the m x n matrix with current element xi and the vector ym as the vector with element yt. The m-vector S of residuals <5, is... 

We now lump the n Lagrange multipliers into the vector X, and the g3(x) into a vector g(x). We further define the m x n matrix F of partial derivatives by its current term fp which is the derivative of the y th constraint with respect to the ith parameter, as... 

Technique 2 Elgenanalysls. It Is well known that the structure of a data set can be uncovered by performing an elgenanalysls of Its covariance matrix.(14) This Is often called principal component analysis. That Is, we arrange the M measurement made on each of N objects as a column vector and combine them to form an M x N matrix, A. A matrix B, resembling the covariance matrix of this data set, Is an M x M matrix AA whose elements are given by... 

Sensitivity analysis has the potential of generating an enormous (perhaps overwhelming) amount of data. The sensitivity coefficients are an m x n matrix, where m is the number of dependent variables in the underlying problem and n is the number of parameters in the analysis. Moreover the sensitivity matrix varies as a function of time. Even for modest-sized problems it is essential to have software and graphics that automatically assist processing and interpreting the sensitivity coefficients. 

A valuable concept in matrix algebra is the hermitean conjugate of a matrix If A is an n x m matrix, then A1 is an m x n matrix with elements... 

M x N matrix, known as the stoichiometric matrix [31, 32], The S,7 entries of the stoichiometric matrix are determined by the stoichiometric numbers appearing in the reactions in the network. For example, if the j 111 reaction has the form ... 

Because the vector m is constrained by the mass conservation requirement = const, the space of possible m values has — 1 dimensions. If the number of independent chemical reactions, R, is less than - 1, then some vectors m are not accessible at some assigned M this, as will be seen, has important consequences in the consideration of heterogeneous chemical equilibria. Now consider the special case where R = N — 1, so that indeed all admissible m s are accessible. Because the kernel of a contains only the zero vector, there exists an M X N matrix A such that... 

Matrices A and B can be multiplied provided that they are conformable in the order written, that is, that the number of columns column order) of the first matrix equals the number of rows row order) of the second. The product C of a m x n matrix A by a n x p matrix B has the elements... 

The inverse mapping from the c space to the c space is equally important as the forward mapping, not only because it provides a link between the lumped species and the original species, but because its existence is a necessary and sufficient condition for exact lumping. For a reduced system hstandard definition of an inverse will not apply. Therefore, we use the concept of a generalized inverse. The generalized inverse of an m X n matrix A satisfies the following criteria ... 

Spreadsheet calculations lend themselves almost automatically to the use of arrays of values. As you ve seen, arrays in Excel can be either one- or two-dimensional. For the solution of many types of problem, it is convenient to manipulate an entire rectangular array of values as a unit. Such an array is termed a matrix. (In Excel, the terms "range"," array " and "matrix" are virtually interchangeable.) An m x n matrix (m rows and n columns) of values is illustrated on the following page. 

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