The n-dimensional cross product and its application to the matrix eigenanalysis
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© 1996 by the author. Published by the American Institute of Aeronautics and Astronautics, Inc. This paper extends to the n-Dimensional space the concept of the cross product by a definition which is derived from the mathematical expression of the row expansion of the matrix determinant. This definition is applied to the eigenvectors computation for a general real matrix having eigenvalues with any algebraic multiplicity. The resulting eigenvector matrix is optimal in terms of condition number. Then an “equivalent” symmetric matrix, which has the same set of eigenvectors, is introduced and a numerical example is included. The application is also extended to the defective matrices having geometrical multiplicity equal to one in order to compute the generalized eigenvectors associated to a true one as well as the upper-diagonal coupling elements. Then the complex case is analyzed and a no-optimal solution is found for it. Finally, the expression of the skew-symmetric “tilde”-matrix performing the cross product in n-Dimensional space is provided.
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