v2006.03.09 - Convex Optimization

478 APPENDIX B. SIMPLE MATRICESB.5.4Matrix rotationOrthogonal matrices are also employed to rotate/reflect like vectors othermatrices: [sic] [88,12.4.1] Given orthogonal matrix Q , the product Q T Awill rotate A ∈ R n×n in the Euclidean sense in R n2 because the Frobeniusnorm is orthogonally invariant (2.2.1);‖Q T A‖ F = √ tr(A T QQ T A) = ‖A‖ F (1256)(likewise for AQ). Were A symmetric, such a rotation would depart fromS n . One remedy is to instead form the product Q T AQ because‖Q T AQ‖ F =√tr(Q T A T QQ T AQ) = ‖A‖ F (1257)Matrix A is orthogonally equivalent to B if B=S T AS for someorthogonal matrix S . Every square matrix, for example, is orthogonallyequivalent to a matrix having equal entries along the main diagonal.[125,2.2, prob.3]B.5.4.1bijectionAny product AQ of orthogonal matrices remains orthogonal. Givenany other dimensionally compatible orthogonal matrix U , the mappingg(A)= U T AQ is a linear bijection on the domain of orthogonal matrices.[148,2.1]