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Unformatted text preview: ENEE 241 02* READING ASSIGNMENT 11 Thu 03/12 Lecture Topics: orthogonal projections Textbook References: 2.12, 2.13 Key Points: • If the columns of V are pairwise orthogonal, then the inner product matrix V H V is diagonal; projection onto the range of V amounts simply to projecting onto each column, then summing the projections. • A m × m matrix V consisting of nonzero ( negationslash = ) pairwise orthogonal vectors is nonsingular. Any vector s in C m can be expressed as s = Vc , where c is found by projecting s onto each of the columns of V . Theory and Examples: 1 . If every pair of reference vectors in V is orthogonal, then the inner product matrix V H V is diagonal, with bardbl v ( j ) bardbl 2 in the ( j,j ) th position. As a result, the solution of the n × n system V H Vc = V H s is simply c j = ( v ( j ) , s ) bardbl v ( j ) bardbl 2 (1 ≤ j ≤ n ) The same c j is used for projecting s onto v ( j ) . Thus Vc is the sum of the projections of s onto each one of the reference vectors. 2 . If the columns of V are (pairwise) orthogonal and nonzero (i.e., negationslash = ), then they are also linearly independent. To see why this is true, recall that linear independence is defined by the property that the equation Vc = has no solution other than c = . Since this equation implies V H Vc = and, in this case, V H V is diagonal with nonzero entries on the leading diagonal, it follows...
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This note was uploaded on 10/18/2011 for the course ENEE 241 taught by Professor Staff during the Spring '08 term at Maryland.
- Spring '08