Rank of the matrix

The degree of the least equation, k, is called the rank of the matrix A. The degree k is never greater than n for the least equation (although there are other equations satisfied by A for which k > n). If A = n, the size of a square matrix, the inverse A exists. If the matrix is not square or k < n, then A has no inverse. 

Matrix methods, in particiilar finding the rank of the matrix, can be used to find the number of independent reactions in a reaction set. If the stoichiometric numbers for the reactions and molecules are put in the form of a matrix, the rank of the matrix gives the number or independent reactions (see Ref. 13). 

Here m < 5, n = 8, p > 3. Choose D, V, i, k, and as the primary variables. By examining the 5x5 matrix associated with those variables, we can see that its determinant is not zero, so the rank of the matrix is m = 5 thus, p = 3. These variables are thus a possible basis set. The dimensions of the other three variables h, p, and Cp must be defined in terms of the primary variables. This can be done by inspection, although linear algebra can be used, too. 

Thus we have illustrated that the number of independent rows, the number of independent columns and the rank of the matrix are all identical. Hence, from geometrical considerations, we conclude that the ranks of the patterns in row- and column-space must also be equal. The above illustration is also rendered geometrically in Fig. 29.7. 

Nf > 0 The problem is underdetermined. If NF > 0, then more process variables exist in the problem than independent equations. The process model is said to be underdetermined, so at least one variable can be optimized. For linear models, the rank of the matrix formed by the coefficients indicates the number of independent equations (see Appendix A). 

Although there is a considerable degree of freedom in the sequence of calculations, when reducing a matrix to a column-echelon form, this is unique and the rank of the matrix is equal to the number of nonzero columns in the column echelon. 

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... 

These messages indicate that the rank of the matrix F is not 5, as expected for the case of a fourth order polynomial, but only 4. Why is this the case ... 

In a casual way one could state that the rank of the matrix equals the number of different species that exist in the mixture. However, such a statement is not generally true and needs to be qualified in several ways ... 

A first question in model-free analysis is how many components are there in a system Or, in other words, what is the rank of the matrix Y In particular, what is the influence of noise Providing an answer to these questions is a first, extremely powerful result of SVD. 

Real data are never noise-free and in purely mathematical terms, the rank of a noisy data matrix is always the smaller of the number of rows or columns. So, the question obviously is, where do we stop What is the correct number of independent species or the correct rank of the matrix Y How many singular values are statistically relevant Most importantly for the chemist what is the practical or the chemical rank how many components are there in the system ... 

The argument can be turned around. If mean-centring reduces the rank of the matrix by one, the data set is closed. 

Oc,. For such a matrix,// -, to have a doubly degenerate root o the rank of the matrix Hy - Eq Sy) must be N - 2, that is, all its first minors must be zero 36). Longuet-Higgins then shows that for a symmetric matrix sufficient condition for this is that Eq be a root of each of the equations. 

What factor analysis allows initially is a determination of the number of components required to reproduce the adsorbance or data matrix A. Factor analysis allows us to find the rank of the matrix A and the rank of A can be interpreted as being equal to the number of absorbing components. To find the rank of A, the matrix ATA is... 

The question is in the data from a real experiment, where many radionuclides are measured in many samples collected under a wide variety of conditions, what is the least number of classes of chemical behavior that will describe the observed results to the desired precision Or, in mathematical terms, what is the rank of the matrix A, and what nuclides should be selected to make up the submatrix a Finally, can any physical significance be attached to the combination of coefficients making up the elements of K, and can these elements of K or quantities thus derived be carried over from one event to the next ... 

The essential point of this chapter is that the physics of the problem we wish to address dictates that the rank of the matrix T is small compared to the dimension N of the bound subspace, where N is the rank of H or of Q. The proof that T is of a low rank begins with the explicit form... 

Proof If t is the rank of the matrix (/S ), then, without loss of generality, we can take the first t columns to be independent and express the last (t -t ) in terms of them,... 

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... 

The principal difference is in the initial conditions to Eq. (3.541) and Eq. (3.539). The initial condition to Eq. (3.539) is much more convenient because it automatically reveals the zero elements of the kernel, thus allowing us to decrease the rank of the matrix equations. The reaction under consideration illustrates this point looking at the matrix form of main operators in the basis of the four collective states of reacting pairs (D B, DB, D B, DB). They are... 

The number of independent reactions R can be found simply as the rank of the matrix of stoichiometric coefficients %J with dimension Sx r such that R < r. Different methods can be applied, such as reduction to triangular matrix by Gaussian elimination for small-size matrices, or computer methods for larger problems. 

Now, with the first linear transformation of the rows the so-called GauEian algorithm is carried out (zero-free main diagonal, beneath it zeros). It determines the rank of the matrix. The rank r = 3 is confirmed. 

However, in this procedure, it could happen that a zero-free diagonal does not arise. Before concluding that the rank of the matrix is in fact r < 3, one should examine whether another arrangement of the core matrix makes a zero-free main diagonal possible. 

The following example has been chosen because it impressively demonstrates the scale-invariance of the pi-space. Besides this, in the matrix transformation we will encounter a reduction of the rank r of the matrix. This will enable us to understand why, in the definition of the pi-theorem (section 2.7), it was pointed out that the rank of the matrix does not always equals the number of base dimensions contained in the dimensions of the respective physical quantities. 

Dimensional analysis of this example is associated by a reduction of the rank of the matrix, because the base dimension of mass is only contained in the density, p. From this it does not follow that the density wouldn t be relevant here, but that it is already fully considered in the kinematic viscosity v, which is defined by v = p/p. Therefore... 

This is a homogeneous system of m linear equations in the c unknowns vn > vn > vic Let us denote by c the rank of the matrix of element indices akJ in the constituent formulae. The number, m, of elements is less than or equal to the number, c, of the constituents i.e. m < c. Furthermore, since all constituents may be formed from the m elements, the number, c, of independent constituents is, at the most, equal to m i.e. c less than m if some constituents may be obtained by chemical reaction between other constituents. Thus, c < m < c. 

The statement of Brinkley s criterion is, therefore the number, c, of independent constituents is equal to the rank of the matrix of element indices in the constituent formulae. The number, s, of independent stoichiometries to be written is s = c — c. ... 

This is a homogeneous system of c linear equations in the s unknowns Xj. Let s be the rank of the matrix of stoichiometric coefficients. If the stoichiometric equations are independent, the X, s are all zero, and thus s = s. 

The statement of Jouguet s criterion is, therefore s stoichiometric equations are independent if the rank of the matrix of the stoichiometric coefficients is equal to s. 

This is an homogeneous system of s linear equations in the c unknowns 7i 72, . 7c- The rank of the matrix of stoichiometric coefficients being equal to s > s, it is possible to choose arbitrarily c — s coefficients 7) and to calculate the others from the Cramer system of the remaining equations. Thus, there exist c —s sets of independent solutions for the coefficients jj, i.e. the number of invariants is equal to c —s. As for stoichiometries, any linear combination of invariants is an invariant. 

The number of independent reactions is just the rank of the matrix S, that is, the rank of the largest nonsingular square submatrix of S. In the example above, the second row is the sum of the other two, so that the rank cannot be 3. On the other hand, the four elements in the upper left-hand corner obviously form a nonsingular submatrix, so that the rank is 2. A nonsingular square submatrix s having the rank of S, is selected... 

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