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# R10 - ENEE 241 02 READING ASSIGNMENT 10 Tue 03/10 Lecture...

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ENEE 241 02* READING ASSIGNMENT 10 Tue 03/10 Lecture Topics: solution of the projection (least squares approximation) problem; complex-valued vectors Textbook References: sections 2.10.4–2.10.5, 2.13 Key Points: The projection of a m -dimensional vector s onto the range R ( V ) of a matrix V is the unique vector ˆ s in R ( V ) with the property that s ˆ s is orthogonal to every column of V . It is also the point in R ( V ) closest to s . ˆ s is obtained by solving the n × n system of equations ( V T V ) c = V T s for c , followed by forming the linear combination ˆ s = Vc . The coefficient vector c is unique if the columns of the matrix V are linearly independent. For complex vectors, the inner product and norm are defined by ( v , w ) = m summationdisplay i =1 v * i w i = ( v * ) T w bardbl v bardbl = ( v , v ) 1 / 2 = parenleftBigg m summationdisplay i =1 | v i | 2 parenrightBigg 1 / 2 ( v * ) T is also denoted as v H . The projection of a complex-valued vector s onto the range of a complex-valued matrix V is given by Vc , where ( V H V ) c = V H s Theory and Examples: 1 . We saw that the projection ˆ s of a m -dimensional vector s onto the range R ( V ) of a matrix V is characterized by two equivalent properties: • bardbl s ˆ s bardbl ≤ bardbl s y bardbl for any y ∈ R ( V ), with equality only when y = ˆ s ; s ˆ s is orthogonal to every column of V . Since ˆ s = Vc for some c R n × 1 , the latter property is also expressed as ( V T V ) c = V T s 2 . V T V is a symmetric n × n matrix. Its ( i, j ) th entry equals the inner product ( v ( i ) , v ( j ) ) . Note also that this matrix does not involve s in any way.

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R10 - ENEE 241 02 READING ASSIGNMENT 10 Tue 03/10 Lecture...

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