## Non Homogeneous System Of Linear Equations

General homogeneous system of equations (rectangular coefficient matrix). A linear system of equations (m equations with n unknowns) is called “homogeneous” if the vector of the right side contains only zero elements (zero vector): resp. . The solvability condition for linear systems of equations, according to which a linear system of equations is solvable if the rank of the coefficient matrix is equal to the rank of the coefficient matrix extended by the vector of the right-hand side, is always satisfied for homogeneous systems of equations because an extension of a matrix with a zero column does not change its rank. Therefore, for homogeneous systems of equations:

• A homogeneous system of equations has always the so-called “trivial solution”

x1 = x2 = x3 = … = xn = 0 .

• In addition to the trivial solution, a homogeneous system of equations also has nontrivial solutions if the rank r(A) of the coefficient matrix is smaller than the number of unknowns n. These solutions are not unique, however, because n-r unknowns can be freely chosen.

Remark: If n > m (more unknowns than equations, “wide” coefficient matrix), the condition r(A) < n is always satisfied, because the rank cannot be greater than the smaller of m and n. Thus, in this case, the system of equations always has nontrivial solutions. These statements are of practical importance mainly for the special case described below. Homogeneous system of equations with quadratic coefficient matrix A homogeneous system of equations with a quadratic coefficient matrix A (n equations with n unknowns) has only the trivial solution, if the matrix A is regular. This case is generally of little interest (note the difference with inhomogeneous systems of equations with quadratic coefficient matrix, where the regular matrix case is generally the only case of interest, because then the system has a unique solution).

 A homogeneous linear equation system with quadratic coefficient matrix (n equations with n unknowns) has non-trivial solutions only if the value of at least one unknown xi is different from zero), if the matrix A is singular. However, these solutions are not unique (the number of free parameters corresponds to the defect of the matrix A). Why is this needed? Homogeneous systems of equations with a quadratic matrix of coefficients arise, for example, in technical mechanics in the treatment of stability problems and vibration problems. Thereby the matrix A contains a parameter (critical load, natural frequency, …), which is determined in such a way that A becomes singular and thus non-trivial (and thus technically interesting) solutions of the equation system exist. The two main variants and solution strategies for the homogeneous systems of equations arising in the above problems are demonstrated using the example of the bending vibrations of a mass-loaded beam.

Examples: The homogeneous system of equations has a regular coefficient matrix (on the page “nth order determinants” it is shown that for the determinant of this matrix det(A) = – 21 holds). Therefore it has only the trivial solution x1 = x2 = x3 = = 0 (any solution method, e.g. the Gaussian algorithm, would also give this result). The homogeneous system of equations on the other hand has a singular coefficient matrix (one can easily convince oneself that det(A) = 0, because the third row is equal to the difference of the double of the second row and the first row). One obtains after some elementary transformations (corresponding to the Gaussian algorithm): Here x3 was chosen as the arbitrary unknown, on the right you see a small selection of possible solutions. The last of the given three solution vectors is the normalized variant of the solution vector (vector of length 1). With the additional requirement “normalized solution vector” the solution becomes unique also for this case. If the defect of the coefficient matrix is larger than 1, then several unknowns can be freely chosen. With the additional requirements for normalized and orthogonal solution vectors, the solution set also becomes unique in these cases. The solution space (set of all solutions) of the homogeneous system of equations is called “null space” or “kernel” of the matrix A is called. Accordingly, the set of orthonormal solution vectors is a “basis of the null space”. Calculation of the “null space basis” with Matlab For the calculation of the zero-space basis of a matrix, Matlab offers the function null. This corresponds (as shown above) to the determination of the nontrivial solutions of a homogeneous system of equations (precondition of the existence of such solutions for a square matrix A is that the matrix is singular). The small test program NullTest.m shown on the left demonstrates the use of this function, which also works with rectangular matrices, with two small square matrices. On the right are the results output to the Command Window. The matrix defined from line 3 A is the regular matrix already used in the example above, with which the homogeneous system of equations has only the trivial solution. Accordingly, the function null cannot return a result (“Empty matrix”). The matrix defined from line 8 B is singular with the defect 2. The function null returns two vectors (output NB). With the result of the product in line 13 it is shown that both vectors fulfill the homogeneous system of equations (the unavoidable rounding errors lead to the fact that no pure zero matrix is delivered as result). The product in row 14 shows that the two-column result matrix does indeed have orthonormal columns. Non Homogeneous System Of Linear Equations.