v2007.09.13 - Convex Optimization
v2007.09.13 - Convex Optimization v2007.09.13 - Convex Optimization
530 APPENDIX B. SIMPLE MATRICEShas R(V W )= N(1 T ) and orthonormal columns. [4] We defined threeauxiliary V -matrices: V , V N (711), and V W sharing some attributes listedin Table B.4.4. For example, V can be expressedV = V W V T W = V N V † N(1435)but V T W V W= I means V is an orthogonal projector (1675) andV † W = V T W , ‖V W ‖ 2 = 1 , V T W1 = 0 (1436)B.4.4Auxiliary V -matrix TabledimV rankV R(V ) N(V T ) V T V V V T V V †V N ×N N −1 N(1 T ) R(1) V V V[ ]V N N ×(N −1) N −1 N(1 T 1) R(1) (I + 2 11T 1 N −1 −1T) V2 −1 IV W N ×(N −1) N −1 N(1 T ) R(1) I V VB.4.5More auxiliary matricesMathar shows [189,2] that any elementary matrix (B.3) of the formV M = I − b1 T ∈ R N×N (1437)such that b T 1 = 1 (confer [111,2]), is an auxiliary V -matrix havingR(V T M ) = N(bT ), R(V M ) = N(1 T )N(V M ) = R(b), N(V T M ) = R(1) (1438)Given X ∈ R n×N , the choice b= 1 N 1 (V M=V ) minimizes ‖X(I − b1 T )‖ F .[113,3.2.1]
B.5. ORTHOGONAL MATRIX 531B.5 Orthogonal matrixB.5.1Vector rotationThe property Q −1 = Q T completely defines an orthogonal matrix Q ∈ R n×nemployed to effect vector rotation; [247,2.6,3.4] [249,6.5] [149,2.1]for x ∈ R n ‖Qx‖ = ‖x‖ (1439)The orthogonal matrix Q is a normal matrix further characterized:Q −1 = Q T , ‖Q‖ 2 = 1 (1440)Applying characterization (1440) to Q T we see it too is an orthogonal matrix.Hence the rows and columns of Q respectively form an orthonormal set.All permutation matrices Ξ , for example, are orthogonal matrices. Thelargest magnitude entry of any orthogonal matrix is 1; for each and everyj ∈1... n‖Q(j,:)‖ ∞ ≤ 1(1441)‖Q(:, j)‖ ∞ ≤ 1Each and every eigenvalue of a (real) orthogonal matrix has magnitude 1λ(Q) ∈ C n , |λ(Q)| = 1 (1442)while only the identity matrix can be simultaneously positive definite andorthogonal.A unitary matrix is a complex generalization of the orthogonal matrix.The conjugate transpose defines it: U −1 = U H . An orthogonal matrix issimply a real unitary matrix.B.5.2ReflectionA matrix for pointwise reflection is defined by imposing symmetry uponthe orthogonal matrix; id est, a reflection matrix is completely definedby Q −1 = Q T = Q . The reflection matrix is an orthogonal matrix,characterized:Q T = Q , Q −1 = Q T , ‖Q‖ 2 = 1 (1443)The Householder matrix (B.3.1) is an example of a symmetric orthogonal(reflection) matrix.
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B.5. ORTHOGONAL MATRIX 531B.5 Orthogonal matrixB.5.1Vector rotationThe property Q −1 = Q T completely defines an orthogonal matrix Q ∈ R n×nemployed to effect vector rotation; [247,2.6,3.4] [249,6.5] [149,2.1]for x ∈ R n ‖Qx‖ = ‖x‖ (1439)The orthogonal matrix Q is a normal matrix further characterized:Q −1 = Q T , ‖Q‖ 2 = 1 (1440)Applying characterization (1440) to Q T we see it too is an orthogonal matrix.Hence the rows and columns of Q respectively form an orthonormal set.All permutation matrices Ξ , for example, are orthogonal matrices. Thelargest magnitude entry of any orthogonal matrix is 1; for each and everyj ∈1... n‖Q(j,:)‖ ∞ ≤ 1(1441)‖Q(:, j)‖ ∞ ≤ 1Each and every eigenvalue of a (real) orthogonal matrix has magnitude 1λ(Q) ∈ C n , |λ(Q)| = 1 (1442)while only the identity matrix can be simultaneously positive definite andorthogonal.A unitary matrix is a complex generalization of the orthogonal matrix.The conjugate transpose defines it: U −1 = U H . An orthogonal matrix issimply a real unitary matrix.B.5.2ReflectionA matrix for pointwise reflection is defined by imposing symmetry uponthe orthogonal matrix; id est, a reflection matrix is completely definedby Q −1 = Q T = Q . The reflection matrix is an orthogonal matrix,characterized:Q T = Q , Q −1 = Q T , ‖Q‖ 2 = 1 (1443)The Householder matrix (B.3.1) is an example of a symmetric orthogonal(reflection) matrix.
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