N-dimensional cross product and its application to matrix eigenanalysis

An extension to n-dimensional space of the cross-product concept by means of a definition derived from the Laplace expansion of the determinant is presented. Based on this definition, which evaluates a vector orthogonal to (n - 1) vectors in the n-dimensional space, some techniques for computing the matrix eigenvectors are developed. These algorithms allow one to compute an orthogonal eigenvector set for any eigenvalue algebraic multiplicity and for real or complex nondefective matrices. An algorithm for computing the generalized eigenvectors and the upper-diagonal coupling elements associated with an eigenvalue having a geometrical multiplicity equal to one (full-defective matrices) is also included. Numerical examples to clarify the proposed algorithms have been presented. Finally, the equivalent symmetric S-matrix, which has the same eigenvector set, and the skew-symmetric tilde matrix, which performs the cross product in n-dimensional space, are provided.

N-dimensional cross product and its application to matrix eigenanalysis Academic Article