[6:] 1 Dr. Casquilho is an Assistant Professor in the Department of Chemical and Biological Engineering at IST (Instituto Superior Técnico), Technical University of Lisbon, Portugal. His email address is mcasquilho@ist.utl.pt. MC IST Comp File=DanilevskiiDuraK.doc Proper values and proper vectors MIGUEL CASQUILHO IST, Technical University of Lisbon, 1049-001 Lisboa, Portugal The calculation of the (distinct) proper values and of the proper vectors of a general, real matrix is presented. The characteristic polynomial is calculated by the method of A. M. Danilevskii, with the roots of the polynomial, which are the proper values, calculated by the method of Durand and Kerner. The method then permits the direct calculation of the proper vectors. This methodology is made available on the Internet. Keywords: proper values, proper vectors, characteristic polynomial, Danilevskii. 1. Fundamentals and scope The calculation of the proper values and proper vectors (also extensively known as eigenvalues and eigenvectors) is necessary in innumerable applications. The calculation of these entities is generally a difficult numerical task for matrices of high order, indeed three or more. Here, a method is presented to solve this problem for a real general, i.e., symmetrical or non-symmetrical square matrices, with the exception of identical proper values. For a (real) symmetrical matrix, its proper values are real, but a non-symmetrical matrix can have complex ones. These will, notwithstanding, be conjugate pairs and, for odd matrix order, at least one real. In what follows, no other distinction is made about the nature (real or complex) of the variables involved. The scope of this stud is to obtain the aforementioned entities. We will start with a real matrix, and apply the method of A. M. Danilevskii to obtain its characteristic polynomial. From this polynomial, we calculate its roots, which are the proper values, by the method of Durand and Kerner. From these, the method of Danilevskii directly supplies the proper vectors. 2. The method of A. M. Danilevskii Let A be a user given real square matrix, or order n. The method of A. M. Danilevskii is based on the conversion of A into Frobenius form by a series of similarity transformations. These preserve the proper values and alter the proper values in a simple way. The matrix A will be successively left- and right-multiplied by n – 1 matrices M –1 and M that differ from the identity matrix only in their row n – 1. Supposing, for the moment, that it is a n,n–1 ≠ 0 (the opposite being treated further below), this row is given by ( 29         - - - = - - - - - - 1 , 1 , 1 , 2 1 , 1 , 1 1 1 n n n n n n n n n n n n j n n a a a a a a a M L {1} for j = 1..n. As is known, ( 29 1 1 - - n M , the inverse of the matrix in Eq. {1}, has its row n – 1 given simply by