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CPSC 491

CPSC 491. Xin Liu November 19, 2010. Norm. Norm is a generalization of Euclidean length in vector space Definition. l p -Norms. Definition Examples l 1 -norm l 2 -norm (Euclidean length) l ∞-norm. Matrix norms. Definition

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CPSC 491

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  1. CPSC 491 Xin Liu November 19, 2010

  2. Norm • Norm is a generalization of Euclidean length in vector space • Definition

  3. lp-Norms • Definition • Examples • l1-norm • l2-norm (Euclidean length) • l∞-norm

  4. Matrix norms • Definition • A matrix norm is a function |||| : RmxnR that satisfies • Frobenius norm • Matrix norms induced by vector norms (see next page)

  5. Induced Matrix Norms • Definition • The supremum of ||Ax|| / ||x|| • The “amplification factor” of vector norms • Examples • The matrix norms of A induced by L1, L2, L∞ norms

  6. Induced Matrix Norms

  7. Induced Matrix Norms • The 1-Norm of a matrix A • Write A in terms of its columns • On the sphere ||x||1 = 1, we have • When x = ej, the upper bound is attained. • Therefore,

  8. Properties of Matrix Norms • For (1) norms induced by vector norms and (2) Frobenius norm, we have the following conclusions • Bound of ||AB|| • ||AB|| ≤||A|| ||B|| • Invariance under Unitary Multiplication • If Q is a unitary matrix, we have||QA|| = ||AQ|| = ||A||

  9. Singular Value Decomposition • Definition • A singular value decomposition (SVD) of A is a factorization • The diagonal entries of Σ are nonnegative and in nonincreasing order • Existence and Uniqueness

  10. Geometric interpretation • The image AS of the unit sphere S under any mxn matrix A is a hyperellipse. • The singular values of A σ1, σ2, …, σn, are the lengths of the principal semiaxes of AS • The left singular vectors of A, {u1, u2, …, un} oriented in the directions of the principal semiaxes of AS • The right singular vectors of A, {v1, v2, …, vn} are the pre-images of the principal semiaxes of AS, i.e., Avj = σjuj

  11. Reduced SVD • If m ≥ n, a full-ranked matrix A has rank n, exactly n singular values are nonzero. • We can reduce the full decomposition by removing the zero rows in Σ, and related columns in U •  • AmxnUcapmxnΣcapnxnVnxn

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