v2010.10.26 - Convex Optimization

786 APPENDIX F. NOTATION AND A FEW DEFINITIONS⌊ ⌋ floor function, ⌊x⌋ is greatest integer not exceeding x| | entrywise absolute value of scalars, vectors, and matriceslogdetnatural (or Napierian) logarithmmatrix determinant‖x‖ vector 2-norm or Euclidean norm ‖x‖ 2√n∑‖x‖ l= l |x j | l vector l-norm for l ≥ 1 (convex)j=1n∑|x j | l vector l-norm for 0 ≤ l < 1 (violates3.2 no.3)j=1‖x‖ ∞ = max{|x j | ∀j} infinity-norm‖x‖ 2 2 = x T x = 〈x , x〉 Euclidean norm square‖x‖ 1 = 1 T |x| 1-norm, dual infinity-norm‖x‖ 0 = 1 T |x| 0 (0 0 0) 0-norm (4.5.1), cardinality of vector x (cardx)‖x‖nk‖X‖ 2k-largest norm (3.2.2.1)= sup ‖Xa‖ 2 = σ 1 = √ λ(X T X) 1 (639) matrix 2-norm (spectral norm),‖a‖=1largest singular value, ‖Xa‖ 2 ≤ ‖X‖ 2 ‖a‖ 2 [159, p.56], ‖δ(x)‖ 2 = ‖x‖ ∞‖X‖ ∗ 2 = 1 T σ(X) nuclear norm, dual spectral norm‖X‖ = ‖X‖ F Frobenius’ matrix norm