Matrix algebra homogeneous equations

The reactance matrix K is ai Q. Exact solutions require the matrix to be of rank n0, implying n linearly independent null-vectors as solutions of the homogeneous equations ma = 0. Because this algebraic condition is not satisfied in general by approximate wave functions, a variational method is needed in order to specify in some sense an optimal approximate solution matrix a. 

The fact that chemical reactions are expressed as linear homogeneous equations allows us to exploit the properties of such equations and to use the associated algebraic tools. Specifically, we use elementary row operations to reduce the stoichiometric matrix to a reduced form, using Gaussian elimination. A reduced matrix is defined as a matrix where all the elements below the diagonal (elements 1,1 2,2 3,3 etc.) are zero. The number of nonzero rows in the reduced matrix indicates the number of independent chemical reactions. (A zero row is defined as a row in which all elements are zero.) The nonzero rows in the reduced matrix represent one set of independent chemical reactions (i.e., stoichiometric relations) for the system. 

The application of matrix algebra for solving sets of simultaneous linear equations - homogeneous and inhomogeneous equations. 

The linear variation method is widely used to find approximate molecular wave functions, and matrix algebra gives the most computationally efficient method to solve the equations of the linear variation method. If the functions fu. .., f in the linear variation function 4> = 2 =i c fit are made to be orthonormal, then = Jfffjdr = 8jj, and the homogeneous set of equations (8.55) for the coefficients that minimize the variational integral becomes... 

SI or cgs units). The above matrix is equivalent to two linear homogeneous algebraic equations (the third and sixth equations) and four linear differential equations (the first, second, fourth, and fifth equations) the third equation is... 

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