Page 108 - Thermodynamics of Biochemical Reactions

5.6 Operations of Linear Algebra 103

H 5.6 OPERATIONS OF LINEAR ALGEBRA

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).
Matrix multiplication. The first dimension of a matrix is the number of rows,
and the second dimension of a matrix is the number of columns. In order to
multiply matrix b by matrix a, it is necessary that the second dimension of matrix
a be the same as the first dimension of matrix b. The product ah has the first
dimension of a and the second dimension of 6. In Mathematica this product is
calculated by putting a period between a and h or by using Dot[a, b].

Gaussian reduction. The rows of a matrix can be multiplied by integers and
be added and subtracted to produce zero elements. This can be done to obtain
the matrix in row-reduced canonical form in which there is a identity matrix on
the left. An identity matrix is a square matrix of zeros with ones along the
diagonal. In Mathematicu the row-reduced canonical form of a is obtained by
using RowReduce[a]. If, after row reduction, one of the rows is made up of zeros,
one of the rows is not independent, and should be deleted. If two matrices have
the same row-reduced form, they are equivalent. We say that a matrix is not
unique because it can be written in different forms that are equivalent.
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 x is said to be the null space
of a. In Muthematica a basis for the null space of a can be calculated by use of
NullSpace[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.
Solution of linear equations. A set of linear equations is represented by ax = b.
The solution x can be obtained in Muthematicu by use of LinearSolve[a,b].
Matrix a can be square or rectangular.
Transposition. In Mathematica the Transpose[a] transposes the first two levels
of a. Equations 5.1-14 and 5.1-23 give a matrix and its transpose.
Pansformation 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.
a(new) = m- 'a(o1d) (5.5-1)
The inverse of the transformation matrix m is represented by m-'. In Mathemat-
ica, the inverse of m is calculated with Inverse[m].