then σ is a singular value of A and u and v are corresponding left and right singular vectors, respectively. (For generality it is assumed that the matrices here are complex, although given these results, the analogs for real matrices are obvious.) If, for a given positive singular value, there are exactly t linearly independent corresponding right singular vectors and t linearly independent co...

A Typical Numerical Computing Problem To illustrate the two paradigms, we will use MATLAB to test a hypothesis regarding Girko’s circular law. Girko’s law states that the eigenvalues of a random N-by-N matrix whose elements are drawn from a normal distribution tend to lie inside a circle of radius sqrt(N) for large enough N. Our hypothesis is that Girko’s circular law can be modified to apply t...

In this paper, a special technique is studied by using the orthogonal Chebyshev polynomials to get approximate solutions for singular and hyper-singular integral equations of the first kind. A singular integral equation is converted to a system of algebraic equations based on using special properties of Chebyshev series. The error bounds are also stated for the regular part of approximate solut...