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lecture11 - EECS 275 Matrix Computation Ming-Hsuan Yang...

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EECS 275 Matrix Computation Ming-Hsuan Yang Electrical Engineering and Computer Science University of California at Merced Merced, CA 95344 h ttp://faculty.ucmerced.edu/mhyang Lecture 11 1 / 27
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Overview Gram-Schmidt process QR decomposition Gram-Schmidt triangular orthogonalization 2 / 27
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Reading Chapter 7 and Chapter 8 of Numerical Linear Algebra by Llyod Trefethen and David Bau Chapter 5 of Matrix Computations by Gene Golub and Charles Van Loan Chapter 5 of Matrix Analysis and Applied Linear Algebra by Carl Meyer 3 / 27
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Geometric properties of orthonormal basis Let the columns of U = [ u 1 . . . u n ] are orthonormal, U > U = I If w = U x , then I mapping w = U x is isometric: it preserves distance (as k w k 2 = k z k 2 ) I inner products are preserved: if w 1 = U x 1 and w 2 = U x 2 , then h w 1 , w 2 i = h x 1 , x 2 i I angles are preserved: ( w 1 , w 2 ) = ( x 1 , x 2 ) Multiplication by U preserves inner products, angles, and distances It follows that U - 1 = U > and hence also UU > = I For any x , x = UU > x , i.e., x = n X i =1 ( u > i x ) u i I u > i x is called the component of x in the direction of u i I a = U > x resolves x into the vector of its u i components I x = U a reconstructs x from its u i components I x = U a = n i =1 a i u i is called the expansion of x 4 / 27
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Examples Rotation by θ in IR 2 is given by y = U θ x , U θ = cos θ - sin θ sin θ cos θ since e 1 = [cos θ, sin θ ] > , e 2 = [ - sin θ, cos θ ] > Reflection across line x 2 = x 1 tan( θ/ 2) is given by y = U θ x , U θ = cos θ sin θ sin θ - cos θ since e 1 = [cos θ, sin θ ] > , e 2 = [sin θ, - cos θ ] > θ θ x 1 x 1 x 2 x 2 e 1 e 1 e 2 e 2 rotation reflection can check that U θ and R θ are orthogonal 5 / 27
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Gram-Schmidt process Given independent vectors x 1 , . . . , x n IR m , Gram-Schmidt process finds orthonormal vectors, q 1 , . . . , q n such that span( x 1 , . . . , x r ) = span( q 1 , . . . , q r ) Thus, q 1 , . . . , q r are orthonormal basis for span( x 1 , . . . , x r ) Idea: first orthogonalized each vector w.r.t. previous ones and then normalize result to have unit norm I v 1 = x 1 I q 1 = v 1 k v 1 k (normalize) I v 2 = x 2 - ( q > 1 x 2 ) q 1 (remove q 1 component from x 2 ) I q 2 = v 2 k v 2 k (normalize) I v 3 = x 3 - ( q > 1 x 3 ) q 1 - ( q > 2 x 3 ) q 2 (remove, q 1 , q 2 components) I q 3 = v 3 k v 3 k I etc. Find orthonormal basis incrementally 6 / 27
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Geometric interpretation of Gram-Schmidt process Gram-Schmidt process: I v 1 = x 1 I q 1 = v 1 k v 1 k (normalize) I v 2 = x 2 - ( q > 1 x 2 ) q 1 (remove q 1 component from x 2 ) I q 2 = v 2 k v 2 k (normalize) I v 3 = x 3 - ( q > 1 x 3 ) q 1 - ( q > 2 x 3 ) q 2 (remove, q 1 , q 2 components) I q 3 = v 3 k v 3 k I etc. i , we have x i = ( q > 1 x i ) q 1 + ( q > 2 x i ) q 2 + · · · + ( q > i - 1 x i ) q i - 1 + k v i k q i = r 1 i q 1 + r 2 i q 2 + · · · + r ii q i 7 / 27
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QR decomposition In matrix form, A = QR where A IR m × n , Q IR m × n , R IR n × n : x 1 x 2 · · · x n | {z } = q 1 q 2 · · · q n | {z } r 11 r 12 · · · r 1 n 0 r 22 · · · r 2 n .
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