n-Dimensional Cross Product and its Application to the Matrix Eigenanalysis
Daniele Mortari · Journal of Guidance Control and Dynamics · 1997
This paper presents an extension to n-dimensional space of the cross-product concept by means of a definition derived from the Laplace expansion of the determinant. 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. The proposed algorithms allow to compute an orthogonal eigenvectors set for any eigenvalue algebraic multiplicity and for real or complex non-defective 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 eigenvectors set, as well as the skew-symmetric tilde-matrix, which performs the cross-product in n-dimensional space, are provided.