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Lec06.Factorization-Duality-3DTransforms

# Lec06.Factorization-Duality-3DTransforms - Factorizing...

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Factorizing Transformations Opposite of Concatenation of Transformations Given a transformation matrix, decompose it into a sequence of simpler transformations Example: Question: How to factorize the multiplicative part? Is the factorization unique? Υ Υ Υ Φ Τ ΢ ΢ ΢ Σ Ρ Υ Υ Υ Φ Τ ΢ ΢ ΢ Σ Ρ = Υ Υ Υ Φ Τ ΢ ΢ ΢ Σ Ρ 1 0 0 0 0 1 0 0 1 0 0 1 1 0 0 4 3 2 1 2 1 2 4 3 1 2 1 a a a a b b b a a b a a Υ Φ Τ ΢ Σ Ρ 4 3 2 1 a a a a

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Special Case: A is symmetric Eigen values of a real symmetric matrix are real Its eigenvectors can always be written as orthonormal matrix A Φ = ΦΛ A = ΦΛΦ T (Implication?) A non symmetric real matrix M can be decomposed as M = U S V T (with U and V being orthonormal, S being a diagonal To compute U S and V, Let A = MM T A = (USV T )(USV T ) T A = US 2 U T V T = (US) -1 M
Singular Value Decomposition Let M be a m -by- n matrix whose entries are real numbers. Then M may be decomposed as M = U S V T where: U is an m -by- m orthonormal matrix S is an m -by- n matrix with non-negative numbers on the main diagonal and zeros elsewhere V is an n -by- n orthonormal matrix Example http://en.wikipedia.org/wiki/Singular_value_decomposition

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