Null space of matrix

Equation 2.67 in Exercise 2.8 is begging to be analyzed by the fundamental theorem of linear algebra [7], so we explore that concept here. Consider an arbitrary m x n matrix, B. The null space of matrix B, written 3 B), is defined to be the set of all vectors x such that S x = 0. The dimension of SEiB) is the. number of linearly independent vectors x satisfying B x == 0. One of the remarkable results of the fundamental theorem of linear algebra is the relation... 

The numerical support for computing null spaces is excellent. For example, the Octave command null CB) returns a matrix with columns consisting of linearly independent vectors in the null space of matrix B. ... 

Calculate the null space of matrix A—null(A)—for the following matrices ... 

In terms of linear vector space, Buckingham s theorem (Theorem 2) simply states that the null space of the dimensional matrix has a fixed dimension, and Van Driest s rule (Theorem 3) then specifies the nullity of the dimensional matrix. The problem of finding a complete set of B-numbers is equivalent to that of computing a fundamental system of solutions of equation 13 called a complete set of B-vectors. For simplicity, the matrix formed by a complete set of B-vectors will be called a complete B-matrix. It can also be demonstrated that the choice of reference dimensions does not affect the B-numbers (22). 

Siace the columns of any complete B-matrix are a basis for the null space of the dimensional matrix, it follows that any two complete B-matrices are related by a nonsingular transformation. In other words, a complete B-matrix itself contains enough information as to which linear combiaations should be formed to obtain the optimized ones. Based on this observation, an efficient algorithm for the generation of an optimized complete B-matrix has been presented (22). No attempt is made here to demonstrate the algorithm. Instead, an example is being used to illustrate the results. 

If the matrix Q is positive semidefinite (positive definite) when projected into the null space of the active constraints, then (3-98) is (strictly) convex and the QP is a global (and unique) minimum. Otherwise, local solutions exist for (3-98), and more extensive global optimization methods are needed to obtain the global solution. Like LPs, convex QPs can be solved in a finite number of steps. However, as seen in Fig. 3-57, these optimal solutions can lie on a vertex, on a constraint boundary, or in the interior. A number of active set strategies have been created that solve the KKT conditions of the QP and incorporate efficient updates of active constraints. Popular methods include null space algorithms, range space methods, and Schur complement methods. As with LPs, QP problems can also be solved with interior point methods [see Wright (1996)]. 

Remark 3. The Q 2 matrix in the Q-R factorization of A2 is such that its columns span the null space of A2. That is,... 

Next we want to consider the fact that a stoichiometric number matrix can be calculated from the conservation matrix, and vice versa. Since Av = 0, the A matrix can be used to calculate a basis for the stoichiometric number matrix v. The stoichiometric number matrix v is referred to as the null space of the A matrix. When the conservation matrix has been row reduced it is in the form A = [/C,Z], where /c is an identity matrix with rank C. A basis for the null space is given by... 

Null space. If the product of two matrices is a zero matrix (all zeros), ax = 0 is said to be a homogeneous equation. The matrix jc is said to be the null space of a. Tn Mathematica a basis for the null space of a can be calculated by use of Null Space [a]. There is a degree of arbitrariness in the null space in that it provides a basis, and alternative forms can be calculated from it, that are equivalent. See Equation 5.1-19 for a method to calculate a basis for the null space by hand. When a basis for the null space of a matrix needs to be compared with another matrix of the same dimensions, they are both row reduced. If the two matrices have the same row-reduced form, they are equivalent. 

Conservation matrix A that corresponds to this stoichiometric matrix is obtained by calculating the null space of (v )T, as indicated by equation 6.3-4. In order to obtain a conservation matrix with identifiable rows, RowReduce is used again and the result is shown in Fig. 6.2. The figure shows that Glu, ATP, ADP, NAD0X, NADred, and Pj can be taken as the six components for glycolysis. This... 

More generally, if R is a matrix that contains a basis for the right null space of (i.e., SR = 0), then... 

Therefore the above sign pattern is not thermodynamically feasible because it is not orthogonal to the sign pattern of a vector from the right null space of the stoichiometric matrix of internal reactions. 

Famili, I. and Palsson, B.O. (2003) The convex basis of the left null space of the stoichiometric matrix leads to the definition of metabolically meaningful pools. Biophys. J. 85, 16-26. 

CxN) NxR) = CxR. Equation 7.1-8 is useful because it makes it possible to calculate a stoichiometric matrix from a conservation matrix. This operation is called taking the null space of A, and the Mathematica operation for doing this is called NullSpace. The use of NullSpace yields a basis for the stoichiometric number matrix. We will see what this means and how it is handled. 

An alternative to the method of Lagrange multipliers for imposing the necessary constraints is sketched below. It derives a lower dimensional unconstrained problem from the original constrained problem by using an orthogonal basis for the null space of the constraint matrix. This method is well suited to the potentially rank-deficient problem at hand, where steps may be taken to... 

Another interesting, extension is the case of larger sets o.f reactions. The equilibrium assumption for larger sets of reactions is elegantly handled by finding the null space of the stoichiometric matrix of the... 

N Null space of the stoichiometric coefficient matrix A N = null(AT)... 

Here, mattix A is the stoichiometric coefficient matrix formed from the reaction stoichiometry, and C is any concentration vector that satisfies Equation 6.7. Determination of the subspace of concenttations orthogonal to the stoichiometric subspace, is in fact equivalent to computing the mil space of A. We shall denote by N the mattix whose columns form a basis for the null space of A. Linear combination of the columns in N hence generates the set of concentrations orthogonal to the stoichiometric subspace. 

Assume that A contains n rows (there are n components in the system). If A has d linearly independent columns (indicating d linearly independent reactions), then rank(A) = d. If the rank of A is d, then the null space of A must have rank (n - d), or rank(N) = (n - d) (Strang, 2003). Thus, N is a matrix having n rows and (n-d) columns. 

Computing the null space of a matrix is common function in linear algebra. We provide a brief description here for convenience however, many standard texts (Lay, 2012 Strang, 2003) describe this topic in detail. 

Since there are more equations than unknowns, Xj = 0 is the only value that produces the zero vector on the right hand side (Axj = 0), and thus the only element in the null space of A is Xj = 0. This is the trivial solution zero will always be part of the null space of any matrix, for any matrix A multiplied by zero will always result in zero on the right hand side. If we let N denote the set of values belonging to the null space of A, N = nuU(A), then for this system N = 0 only. 

Determine the matrix N—the matrix with columns that form a basis for the null space of A. ... 

Notice that if we substitute matrix X discussed in this example with the stoichiometric coefficient matrix A, then the columns in X (Xj and X2) represent two reactions participating in n-dimensional concentration space, R". Hence, to compute N, we simply determine the stoichiometric coefficient matrix A (as in Section 6.2.1.3), and then compute the null space of A. From linear algebra, we can show that if A has size nxd (n components participating in d reactions), the size of matrix N will benx(n-d). 

A is a 3 X 5 matrix and thus the null space of A will be a two-dimensional subspace in c -Cb-Cc-Cd-Ce space (the size of matrix N must he nx(n- d), or 5x2). To compute the null space of this matrix, we can reduce A to reduced row echelon form by performing elementary row operations on A, and determine all of the vectors in the null space (similar in procedure to that shown in Example 3). Hence reducing A to the equivalent matrix gives ... 

A(C) is found from the controllability matrix E for the CSTR. To construct E, the set of vectors orthogonal to the stoichiometric subspace must be known. This is done by finding a basis for the null space of the vectors spanned by the reaction system given by the stoichiometric coefficient matrix A. 

An alternative method for computing S exists that relies on computing a basis for the null space of the stoichiometric matrix A. Vectors belonging to null (A ) are normal to S, and these vectors may be related to bounding hyperplane constraints that define S. This method is described in Feinberg (1987,2000a, 2000b). 

They have proved that if we have a matrix M with the above mentioned conditions than the null space of M (the eigenvectors c, c and c of the eigenvalue X = 0 ) gives a proper embedding of G(V, E) in the sphere S as x,- = c + for i = 1,..., 3, and xi p = 1. Thus the relation (xu,yu,Zu) = (x(u)j,x(u)2,x(u)3) is valid for each vertex. It was also proved that this null space contains bi-lobal eigenvectors (van der Holst 1996). The matrix M is often called Colin de Verdiere matrix in the scientific literature. 

GetMatr ixN returns the null space of the system matrix. 

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