Linear matrix rows

The degree of the least polynomial of a square matr ix A, and henee its rank, is the number of linearly independent rows in A. A linearly independent row of A is a row that eannot be obtained from any other row in A by multiplieation by a number. If matrix A has, as its elements, the eoeffieients of a set of simultaneous nonhomo-geneous equations, the rank k is the number of independent equations. If A = , there are the same number of independent equations as unknowns A has an inverse and a unique solution set exists. If k < n, the number of independent equations is less than the number of unknowns A does not have an inverse and no unique solution set exists. The matrix A is square, henee k > n is not possible. 

The process design linear program model is best written with flexibility in mind, such as extra matrix rows to provide flexibility in recycling, adding outside streams intermediate in the process, and determining component incremental values at each processing stage. This subject is discussed more fully later in this chapter. 

Rank of a matrix The rank of a matrix is equal to the number of linearly independent rows or eolumns. The rank ean be found by determining the largest square... 

As mentioned earlier, singular matrices have a determinant of zero value. This outcome occurs when a row or column contains all zeros or when a row (or column) in the matrix is linearly dependent on one or more of the other rows (or columns). It can be shown that for a square matrix, row dependence implies column dependence. By definition the columns of A, a, are linearly independent if... 

The need to solve sets of linear equations arises in many optimization applications. Consider Equation (A.20), where A is an n X n matrix corresponding to the coefficients in n equations in n unknowns. Because x = A 1b, then from (A.17) A must be nonzero A must have rank w, that is, no linearly dependent rows or columns exist, for a unique solution. Let us illustrate two cases where A = 0 ... 

The stoichiometric matrix N consists of m rows, corresponding to m metabolic reactants, and r columns, corresponding to r biochemical reactions or transport processes (see Fig. 5 for an example). Within a metabolic network, the number of reactions (columns) is usually of the same order of magnitude as the number of metabolites (rows), typically with slightly more reactions than metabolites [138]. Due to conservation relationships, giving rise to linearly dependent rows in N, the stoichiometric matrix is usually not of full rank, but... 

Note that the E is not unique, each nonsingular linear transformation is again a valid representation of the left nullspace. The matrix E consists of in rank ( A7) rows, corresponding to mass-conservation relationships (and a linearly dependent rows) in N. In particular,... 

Denoting with N° the matrix that consists only of the first rank(A) linearly independent rows of N (corresponding to the independent species Sinci ), the full set of differential equations is given as... 

As a simple example, consider the minimal glycolytic pathway shown in Fig. 5. The stoichiometric matrix N has m = 5 rows (metabolites) and m = 6 columns (reactions and transport processes). The rank of the matrix is rank(A) = 4, corresponding to m — ran l< (TVj = 1 linearly dependent row in N. The left nullspace E can be written as... 

Within the Matlab s numerical precision X is singular, i.e. the two rows (and columns) are identical, and this represents the simplest form of linear dependence. In this context, it is convenient to introduce the rank of a matrix as the number of linearly independent rows (and columns). If the rank of a square matrix is less than its dimensions then the matrix is call rank-deficient and singular. In the latter example, rank(X)=l, and less than the dimensions of X. Thus, matrix inversion is impossible due to singularity, while, in the former example, matrix X must have had full rank. Matlab provides the function rank in order to test for the rank of a matrix. For more information on this topic see Chapter 2.2, Solving Systems of Linear Equations, the Matlab manuals or any textbook on linear algebra. 

The rank of a matrix Y is the number of linearly independent rows or columns in this matrix. The columns of Y are linearly dependent if one of the column vectors y j can be written as a linear combination of the other columns. The same holds for rows. 

In this section we review the known theorems that relate entanglement to the ranks of density matrices [52]. The rank of a matrix p, denoted as rank(p), is the maximal number of linearly independent row vectors (also column vectors) in the matrix p. Based on the ranks of reduced density matrices, one can derive necessary conditions for the separability of multiparticle arbitrary-dimensional mixed states, which are equivalent to sufficient conditions for entanglement [53]. For convenience, let us introduce the following definitions [54—56]. A pure state p of N particles Ai, A2,..., is called entangled when it cannot be written... 

Horiuti numbers v (S x P) is the route of a complex reaction. The rank of the matrix rint cannot be higher than (S - P) since, according to eqn. (19) there are P linearly independent rows of Tint. As usual, we have... 

Assumption 6.4. There exist a full column rank matrix B(x, 6) B C JrAx.Y-1 and a matrix f (x,0) C Jt(X-1)xml with linearly independent rows, such that r (x, 0) can be rewritten as... 

Similarly, the number of linearly independent rows of A is called the row-rank of A. The row-rank of A is the column-rank of A. A fundamental theorem in matrix algebra states that the row-rank and the column-rank of a matrix are equal (and equal to the rank) [Schott 1997], Hence, it follows that the rank r(A) < min(/,/). The matrix A has full rank if and only if r(A) = mini/,/). Sometimes the term full column-rank is used. This means that r(A) = min(/,/) =. /, implying that J < I. The term full row-rank is defined analogously. 

The size of a matrix is the number of rows and columns hence, X in Eq. (A.4) is 4 x 5 (4 rows by 5 columns). The rank of a matrix is the number of linearly independent rows or columns (since the row and column rank of a matrix are the same). A matrix of size n x p that has rank less than p cannot be inverted (matrix inversion is discussed later). Similarly, a matrix of size n x n cannot be inverted if its rank is less than n. 

In equations 7.13,7.25, and 7.27 we have denoted L+ = (L Ly L1, and similarly for Mi+ in equation 7.27, and D+ in equation 7.28, and N4- in equation 7.29. This notation is used because these are all special cases of the Moor e-Penrose Pseudoinverse M which can be defined for an arbitrary matrix M and which gives the minimum least-squares approximation even in cases where the columns of M may not be linearly independent (see Lawson and Hanson, 1974). Similarly, in equations 7.25, 7.27, and 7.29 we have denoted Ai+ = A/ (AiA/) 1 since this is another special case of the Moore-Penrose Pseudoinverse, for the case where the matrix in question, Ai, has linearly independent rows. 

Obviously, if Hp = K Eq. (4) converges at the first iteration and is fully identical to Eq. (3). An important advantage of iterative techniques is that iterations are stable even if Eq. (2) does not have a unique solution. Indeed, if the matrix K has linearly dependent rows, applying Eq. (2) is problematic, since matrix K does not exist. Iterating Eq. (4) works even in such a case, with the difference that use of Eq. (4) leads to one of many possible solutions that depend on initial guess. 

In terms of considerations given in Section 3, ill-posed problems have a non-unique and/or an unstable solution. For a non-unique solution, the matrix K C K on the left side of Eq. (11c) has linearly dependent rows (and columns, since it is symmetrical), i.e. det(K C K)=0 (degenerated matrix) and the inverse operator (K C K)" does not exist. For a quasi-degenerated matrix (det(K C K) 0) the inverse operator (K C K)" exists. However, in... 

R in the DMRG representation. This allows terminating the Gram-Schmidt orthonormalization process when the number of orthonormal functions obttiined is Ir- In prax tice, we first obtain the matrix representation of in the direct product DMRG basis in a way which is analogous to the setting-up of the full Hamiltonian matrix from the blocks. The linear dependencies discussed above manifest as linearly dependent rows of the matrix of Pr. 

The rank of a matrix is the maximum number of linearly independent vectors (rows or columns) in an X/t matrix denoted as r X). Linearly dependent rows or columns reduce the rank of a matrix. 

The number of inequality constraints, /, can be greater than m, the number of controls. That the number of active constraints at any time does not exceed the m is assured by the constraint qualification. It requires that if p inequality constraints are active (i.e., / = 0 for f = 1,2,..., p), then p should be the rank of the matrix of partial derivatives of f with respect to u. Note that p is the number of linearly independent rows or columns of the matrix (see Section 4.3.2.1, p. 97). 

B.2.1.8 Row and Column Space The column space of a matrix A is the vector space generated by all linear combinations of the column vectors of A. Hence, the column space of A is equal to the span of the columns of A. Similarly, the row space of matrix A is the vector space that is generated by all combinations of the row vectors of A. The dimension of the column space is thus equal to the number of linearly independent column vectors in A, whereas the dimension of the row space of A is equal to the number of linearly independent row vectors in A, which are both equal to the rank of A, rank(A). Hence, the dimension of the column space of... 

Because n e some solution in W, exists. Clearly, it is uniquely determined if and only if the matrix (v, ,) is of full column rank R, which means R = Rq = dimi otherwise there is an infinity of such solutions. This is the case oi linearly independent reactions. If we then take arbitrary R linearly independent rows of the matrix, thus the corresponding R (= Rq< K) equations (4.5.12), the integral reaction rates can be computed. So when desired, we have also obtained an information on how the reaction kinetics works in the reaction node. If the term is further integrated over an interval of time (say, from r, to t2), the integral is called the extent of the r-th reaction from initial r, to time tj the concept is useful rather and in particular for batch reactors. 

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