Null space

In our companion book (Vol. 1 - Buzzi-Ferraris and Manenti, 2010a), we introduced readers to several criteria for finding the null space N of a matrix A e m n. If the rank of A is m, N  

The null space N of a matrix A is a matrix such that 

Since the matrix N may be of interest, the function GetBzzMatrixN was implemented in the BzzFactorizedLQ class of the BzzMoth library. 

A second alternative to evaluate the null space, N G pnx n m) matrix A is as follows. 

The variables x are divided into dependent and independent variables. The m dependent variables are denoted by and the remaining n — m independent 

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The first requirement is the definition of a low-dimensional space of reaction coordinates that still captures the essential dynamics of the processes we consider. Motions in the perpendicular null space should have irrelevant detail and equilibrate fast, preferably on a time scale that is separated from the time scale of the essential motions. Motions in the two spaces are separated much like is done in the Born-Oppenheimer approximation. The average influence of the fast motions on the essential degrees of freedom must be taken into account this concerns (i) correlations with positions expressed in a potential of mean force, (ii) correlations with velocities expressed in frictional terms, and iit) an uncorrelated remainder that can be modeled by stochastic terms. Of course, this scheme is the general idea behind the well-known Langevin and Brownian dynamics. 

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)]. 

The columns of P span the null space of A2, and thus the unmeasured variables are eliminated. 

The critical step of the inverse problem is a projection of the unknown function onto the base formed by the kernel functions. This base is not orthogonal and, because the number of kernel functions is finite, cannot describe perfectly the unknown function. For that reason, we write that the unknown function x /SC x) is the sum of its projections with component u, along the j th kernel functions plus any orthogonal complement labeled x fiC x) and belonging to the null-space of the kernel functions... 

Starting with an evaluation of the stoichiometric matrix, we obtain the null space matrix K and the link matrix L,... 

There are situations when it can be difficult to obtain a sufficiently independent set of RDC measurements, such as in small molecules or in cases when poor chemical shift resolution limits the number of data points. As pointed out by Losonzci et al.,S9 in these cases it is still possible to restrict the possible solutions for the order tensor, with remaining ambiguities possibly lifted using additional experimental means or physical considerations. The procedure for dealing with this underdetermined case is to generate a unique solution for the existing data and then supplement this solution with vectors drawn randomly from the so-called null space. All solutions constructed in this manner are equally valid. The full set of solutions for the best fit order tensor in this underdetermined case is written,50... 

Figure 2. CIO volume mixing ratio profiles across the vortex edge as measured by the ASUR instrument (solid line) on March 6, 1996, compared to results of the SI.IMCAT 3-D model (dashed line, rectangles) for identical positions at 12 UT. Also indicated are average times, positions, solar zenith angles, and 475 K-PV s for the individual measurements. The thick error bars represent the relevant statistical errors due to measurement noise, the thin dotted error bars include also the error due to the limited altitude resolution (the so-called null-space error). Modelled and measured data are in relatively good agreement inside, at the edge, and outside the Arctic polar vortex.

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... 

Mathematica is very useful for carrying out these matrix operations. The operation for row reduction is RowReduce, and the operation for calculating a basis for the null space is NullSpace. Row reduction is also used to determine whether the equations in a set of conservation equations or reaction equations are independent. Rows that are dependent come out as all zeros when this is done, and they must be deleted because they do not provide any useful information. 

A basis for the null space v of conservation matrix 5.2-5 at specified pH obtained with equation 5.1-19 or with a computer is... 

Although matrix multiplications, row reductions, and calculation of null spaces can be done by hand for small matrices, a computer with programs for linear algebra are needed for large matrices. Mathematica is very convenient for this purpose. More information about the operations of linear algebra can be obtained from textbooks (Strang, 1988), but this section provides a brief introduction to making calculations with Mathematica (Wolfram, 1999). 

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. 

Transformation matrix. When the conservation matrix a for a system is written in terms of elemental compositions, the elements are used as components. But we can change the choice of components (change the basis) by making a matrix multiplication that does not change the row-reduced form of the a matrix or its null space. Since components are really coordinates, we can shift to a new coordinate system by multiplying by the inverse of the transformation matrix between the two coordinate systems. A new choice of components can be made by use of a component transformation matrix m, which gives the composition of the new components (columns) in terms of the old components (rows). The following matrix multiplication yields a new a matrix in terms of the new components. 

The apparent stoichiometric number matrix v" can be obtained from the row-reduced form of A" by use of the analogue of equation 5.1-19 or by calculating a basis for the null space using a computer program. 

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... 

Here, the matrix S of Equation (9.4) has a one-dimensional right null space, for which the vector [1 1 1 r is a basis. Equation (9.22) corresponds to summing the reaction potentials about the closed loop formed by the reactions in Equation (9.3). 

For a stoichiometric matrix associated with the internal reactions of a given system, with right null space R, the vector of internal fluxes J is thermodynamically feasible if and only if... 

All possible sign patterns in the right null space are captured in the set of three cycles 4... 

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. 

This is why calculating the null space yields a basis for the conservation matrix. The row reduced conservation matrix can be labelled as follows ... 

Therefore, the linear inverse problem (10.4) can have an infinite number of equivalent solutions. All these nonradiating currents form a so-called null-space of the linear inverse problem (10.4). In principle, we can overcome this difficulty using a regularization method, which restricts the class of the inverse excess currents j in equation (10.4) to physically meaningful solutions only. We will discuss an approach to the solution of this problem below, considering a quasi-linear method. 

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... 

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