By Kondratiev A. S., Mazurov V. D.

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1) the following Hermitian form of ÷í T (í) with respect to a set of complex variables fxi ; i P g À1 (í)g for each í is positive de®nite, because 2 n (í) Ã ÷í x i T ij x j T ki xi . 0 (2X5X7a) k1 i, jP gÀ1 (í) i where xÃi is the complex conjugate of xi . 7a) are arbitrary as long as the indices i belong to gÀ1 (í), any principal submatrices of ÷í T (í) are also positive de®nite; hence, their eigenvalues are all positive. Thus, their determinants (called the principal minors of ÷í T (í) ) are also positive: the ®rst few of them are ÷í T (í) jT ik j2 .

It will be shown that any real orthogonal matrix of a ®nite order can be expressed in this exponential form. ) 4. 1a) can be written in an exponential form as follows: ! 0 Àè (2) W R e ; W (1X6X2) è 0 Hint. 1e) back to R(2) , using T ó z T y ó y . 5. The vector product [s 3 x] of two vectors s and x in three dimensions is de®ned by the components s2 x3 À s3 x2 , s3 x1 À s1 x3 , s1 x2 À s2 x1 Show that it can be written in the form [s 3 x] ùx where ù is a 3 3 3 antisymmetric matrix de®ned by P Q 0 Às3 s2 ù ù(s) R s3 0 Às1 S Às2 s1 0 (1X6X3a) (1X6X3b) 6.

0; i P g À1 (í) ii 4 2 (÷í ) det k T (í) ii T (í) ij T (í) ji T (í) jj 5 (2X5X7b) . 0; À1 i, j P g (í) and so on. 7b) means that the signs of all diagonal elements of T (í) are given by the sign of ÷í . If any one of the diagonal elements, T ii , is zero, then T becomes singular since the ith column and row become null. 7a), will provide the condition for T (í) to be diagonalized by the Gaussian elimination procedure which will be discussed in the next section. 5) that the complete diagonalizations of the matrix A can be achieved by diagonalizing each submatrix T (í) of order ní 3 ní.