The proof shows that (2.1) is a generalization of (2.2) by replacing the orthogonal matrices
by weighted orthogonal matrices
is, the set of [m.sub.1] x [m.sub.2] column orthogonal matrices
is written as [mathematical expression not reproducible].
The only difference between OTSA and TSA or between ODTSA and DTSA is that U and V are constrained to orthogonal matrices
in OTSA and ODTSA.
For given symmetric orthogonal matrices
[R.sub.3] [member of] [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], [R.sub.4] [member of] [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], [R.sub.5] [member of] [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] and [S.sub.5] [member of] [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], then matrix equation (2) is solvable if and only if the following matrix equations are consistent, namely
[v.sub.N]) can be obtained on the basis of singular value decomposition of the matrix [[PHI].sup.[summation]] (Low et al, 1986): [[PHI].sup.[summation]] = [U.sub.[PHI]] x [V.sub.[PHI]], where [U.sub.[PHI]] and [V.sub.[PHI]] are orthogonal matrices
, [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], and are singular values of [[PHI].sup.[summation]].
However the optimal approximation solution to a given matrix pair ([G.sup.*.sub.1], [G.sup.*.sub.2]) cannot be obtained in the corresponding solution set, and the difficulty is due to that the invariance of the Frobenius norm only holds for orthogonal matrices
, but does not hold non-singular matrices that appear in CCD used in .
Topics include orthogonal matrices
and wireless communications, probabilistic expectations on unstructured spaces, higher order necessary conditions in smooth constrained optimization, Hamiltonian paths and hyperbolic patterns, fair allocation methods for coalition games, sums-of-squares formulas, product-free subsets of groups, generalizations of product- free subsets, and vertex algebras and twisted bialgebras.
For example: Lecturing on "orthogonal matrices
" which were introduced by the French mathematician Charles Hermite in 1854 we went further back to 1770 when Euler for the first time considered a system of linear equations in which an orthogonal matrix was used implicitly without knowing anything about matrices in general or orthogonal matrices
All two dimensional orthogonal matrices
have the following structure:
 Ann Lee: Secondary symmetric, secondary skew symmetric, secondary orthogonal matrices
; Period Math.
As one might gather from this result, the set of all n x n orthogonal matrices
plays a central role in our analysis.
where U (mxm) and V (nxn) are square, orthogonal matrices
and [SIGMA] is a diagonal matrix mxn of singular values ([sigma]i j = 0 if i [not equal to] j and [sigma]11 [greater than or equal to] [sigma]22 [greater than or equal to] ...