Revisit of Fredholm alternative theorem by using SVD and bordered matrix

Abstract

Fredholm alternative theorem plays an important role in the linear algebra when the matrix is singular. Base on the singular value decomposition (SVD) for the matrix, the null space and range deficiency can be easily and systematically understood. By introducing a slack variable, we obtain a bordered matrix by adding one column vector from the left singular vector with respect to the zero singular value and one row vector from the right singular vector with respect to the zero singular value. It is interesting to find that an original singular matrix is regularized to a non-singular one. The value of the slack variable indicates the infinite solution (zero) or no solution (non-zero) for the original linear algebraic system. To demonstrate this finding, one triangular-domain problem with a degenerate scale and a rigid body mode is solved. Although influence matrices are singular in the boundary integral equation formulation for different problems (degenerate scale in the Dirichlet problem and rigid body mode in the Neumann problem), the corresponding unique solution (Dirichlet problem) and infinite solutions containing a constant potential (Neumann problem) can be obtained by using the bordered matrix and SVD technique. Besides, free-free flexibility matrix of finite element method is also derived from the stiffness matrix by a self-regularization technique.