v2010.10.26 - Convex Optimization
v2010.10.26 - Convex Optimization v2010.10.26 - Convex Optimization
650 APPENDIX B. SIMPLE MATRICESConsider rotation of these vertices via orthogonal matrixQ [1 1 √ n+1ΞV W ]Ξ ∈ R n+1×n+1 (1672)where permutation matrix Ξ∈ S n+1 is defined in (1728), and V W ∈ R n+1×nis the orthonormal auxiliary matrix defined inB.4.3. This particularorthogonal matrix is selected because it rotates any point in subspace R nabout one axis of revolution onto R ; e.g., rotation Qe n+1 aligns the laststandard basis vector with subspace normal R ⊥ = 1. The rotated standardbasis vectors remaining are orthonormal spanning R .Another interpretation of product QX is rotation/reflection of R(X).Rotation of X as in QXQ T is a simultaneous rotation/reflection of rangeand rowspace. B.8Proof. Any matrix can be expressed as a singular value decompositionX = UΣW T (1563) where δ 2 (Σ) = Σ , R(U) ⊇ R(X) , and R(W) ⊇ R(X T ).B.5.4Matrix rotationOrthogonal matrices are also employed to rotate/reflect other matrices likevectors: [159,12.4.1] Given orthogonal matrix Q , the product Q T A willrotate A∈ R n×n in the Euclidean sense in R n2 because Frobenius’ norm isorthogonally invariant (2.2.1);‖Q T A‖ F = √ tr(A T QQ T A) = ‖A‖ F (1673)(likewise for AQ). Were A symmetric, such a rotation would depart from S n .One remedy is to instead form product Q T AQ because‖Q T AQ‖ F = √ tr(Q T A T QQ T AQ) = ‖A‖ F (1674)ByA.1.1 no.31,vec Q T AQ = (Q ⊗ Q) T vec A (1675)B.8 The product Q T AQ can be regarded as a coordinate transformation; e.g., givenlinear map y =Ax : R n → R n and orthogonal Q, the transformation Qy =AQx is arotation/reflection of range and rowspace (141) of matrix A where Qy ∈ R(A) andQx∈ R(A T ) (142).
B.5. ORTHOGONAL MATRIX 651which is a rotation of the vectorized A matrix because any Kronecker productof orthogonal matrices remains orthogonal; id est, byA.1.1 no.38,(Q ⊗ Q) T (Q ⊗ Q) = I (1676)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.[202,2.2, prob.3]
- Page 599 and 600: A.3. PROPER STATEMENTS 599(AB) T
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- Page 607 and 608: A.3. PROPER STATEMENTS 607For A,B
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- Page 611 and 612: A.4. SCHUR COMPLEMENT 611A.4 Schur
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- Page 615 and 616: A.4. SCHUR COMPLEMENT 615From Corol
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- Page 625 and 626: A.7. ZEROS 625A.6.5SVD of symmetric
- Page 627 and 628: A.7. ZEROS 627(Transpose.)Likewise,
- Page 629 and 630: A.7. ZEROS 629For X,A∈ S M +[34,2
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- Page 633 and 634: Appendix BSimple matricesMathematic
- Page 635 and 636: B.1. RANK-ONE MATRIX (DYAD) 635R(v)
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- Page 639 and 640: B.2. DOUBLET 639R([u v ])R(Π)= R([
- Page 641 and 642: B.3. ELEMENTARY MATRIX 641has N −
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- Page 647 and 648: B.5. ORTHOGONAL MATRIX 647Given X
- Page 649: B.5. ORTHOGONAL MATRIX 649Figure 15
- Page 653 and 654: Appendix CSome analytical optimal r
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- Page 657 and 658: C.2. TRACE, SINGULAR AND EIGEN VALU
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- Page 669 and 670: Appendix DMatrix calculusFrom too m
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- Page 699 and 700: Appendix EProjectionFor any A∈ R
B.5. ORTHOGONAL MATRIX 651which is a rotation of the vectorized A matrix because any Kronecker productof orthogonal matrices remains orthogonal; id est, byA.1.1 no.38,(Q ⊗ Q) T (Q ⊗ Q) = I (1676)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.[202,2.2, prob.3]