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Unformatted text preview: r equivalently,
Ax − λx = 0.
We also obtain the condition
∂H
= −g (x) = 0,
∂λ
which is just our constraint. Thus the point on the sphere at which f assumes
its maximum is a unitlength eigenvector b1 , the eigenvalue being the value λ1
of the variable λ.
Let W be the “linear subspace” of R n deﬁned by the homogeneous linear
equation b1 · x = 0. The intersection S n−1 ∩ W is a sphere of dimension n − 2.
31 We next use the method of Lagrange multipliers to ﬁnd a point on S n−1 ∩ W
at which f assumes its maximum. To do this, we let
g1 (x) = xT x − 1, g2 (x) = b1 · x. The maximum value of f on S n−1 ∩ W will be assumed at a critical point for
the function
H (x, λ, µ) = f (x) − λg1 (x) − µg2 (x).
This time, diﬀerentiation shows that
∂H
= 2ai1 x1 + 2ai2 x2 + · · · + 2ain xn − 2λxi − µbi = 0,
∂xi
or equivalently,
Ax − λx − µb1 = 0. (2.2) It follows from the constraint equation b1 · x = 0 that
b1 · (Ax) = bT (Ax) = (bT A)x = (bT AT )x
1
1
1
= (Ab1 )T x = (λ1 b1 )T x = λ1 b1 · x = 0.
Hence...
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This document was uploaded on 01/12/2014.
 Winter '14
 Equations

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