Engineering Calculus Notes 385

Engineering Calculus Notes 385 - u 1 P = u 1 = { x R 3 | x...

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3.9. THE PRINCIPAL AXIS THEOREM 373 Equation ( 3.31 ) an eigenvector of A ; the associated scalar λ is called the eigenvalue of A associated to −→ u . 20 Our discussion has shown that every symmetric matrix A has an eigenvector, corresponding to the minimum of the associated quadratic form Q ( −→ x ) = −→ x · A −→ x on the unit sphere. In fact, we can say more: Proposition 3.9.2 (Principal Axis Theorem for R 3 ) . If A is a symmetric 3 × 3 matrix, then there exist three mutually perpendicular unit eigenvectors for A : −→ u i , i = 1 , 2 , 3 satisfying A −→ u i = λ i −→ u i for some scalars λ i R , i = 1 , 2 , 3 , −→ u i · −→ u j = b 0 if i n = j, 1 if i = j. Proof. Since the unit sphere S 2 is sequentially compact, the restriction to S 2 of the quadratic form Q ( −→ x ) = −→ x · A −→ x achieves its minimum somewhere, say −→ u 1 , and this is a solution of the equations A −→ u 1 = λ 1 −→ u 1 (some λ 1 ) −→ u 1 · −→ u 1 = 1 . Now, consider the plane P through the origin perpendicular to
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Unformatted text preview: u 1 P = u 1 = { x R 3 | x u = 0 } and look at the restriction of Q to the circle P S 2 = { u R 3 | u u 1 = 0 and u u = 1 } . This has a minimum at u 2 and a maximum at u 3 , each of which is a solution of the Lagrange multiplier equations f ( u ) = g 1 ( u ) + g 2 ( u ) g 1 ( u ) = u u = 1 g 2 ( u ) = u u 1 = 0 . Again, we have, for i = 1 , 2 , 3, f ( u ) = 2 A u g 1 ( u ) = 2 u 20 Another terminology calls an eigenvector a characteristic vector and an eigenvalue a characteristic value of A ....
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