H freeman new york 1990 see chapter 5 33 the coecient

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Unformatted text preview: it follows from (2.2) that Ax − λx = 0. Thus if b2 is a point on S n−1 ∩ W at which f assumes its maximum, b2 must be a unit-length eigenvector for A which is perpendicular to b1 . Continuing in this way we finally obtain n mutually orthogonal unit-length eigenvectors b1 , b2 , . . . , bn . These eigenvectors satisfy the equations Ab1 = λ1 b1 , Ab2 = λ2 b2 , ... Abn = λ2 bn , which can be put together in a single matrix equation Ab1 Ab2 · Abn = λ1 b1 λ2 b2 · λn bn , or equivalently, A b1 b2 · bn = b1 · bn b2 λ1 0 · 0 If we set B = b1 b2 · bn , this last equation becomes λ1 0 AB = B · 0 0 λ2 · 0 32 ·0 · 0 . · · · λn 0 λ2 · 0 ·0 · 0 . · · · λn Of course, B is an orthogonal matrix, so it is invertible and obtaining λ1 0 · 0 λ1 0 λ2 · 0 −1 0 −1 B , or B AB = A=B · · · · · 0 0 · λn 0 we can solve for A, 0 λ2 · 0 ·0 · 0 . · · · λn We can arrange that the determinant of B be one by changing the sign of one of the columns if necessary....
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