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Unformatted text preview: Some useful linear algebra Linearly independent vectors • • span(V): span of vector space V is all linear combinations of vectors v i, i.e. for only 2 1 3 3 2 2 1 1 = = = = = + + + + i i i v v v v λ λ λ λ λ λ λ i i i v ∑ λ singular is ) ( hence ) ( A I x A I x Ax = → = λ λ λ The eigenvalues of A are the roots of the characteristic equation ) det( ) ( = = A I p λ λ = Λ = N AS S λ λ λ . 2 1 1 Eigenvectors of A are columns of S diagonal form of matrix AM M B 1 If = Similarity transform then A and B have the same eigenvalues The eigenvector x of A corresponds to the eigenvector M1 x of B Rank and Nullspace 1 1 × × × = m n n m b x A Least Squares b Ax = • More equations than unknowns • Look for solution which minimizes Axb = (Axb) T (Axb) • Solve • Same as the solution to • LS solution ) ( ) ( = ∂ ∂ i T x b Ax b Ax b A Ax A T T = b A A A x T T 1 ) ( = Properties of SVD...
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 Winter '08
 staff
 Linear Algebra, ax, linearly independent vectors, AAT Columns, useful linear algebra

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