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3.4730 3.3709 0.5898 2.1181 9.5023 7.7400 10.334 3.0674 0.8371 6.6037 16.798 7.9869 2.9252 0.4791 0.5046 0.0 124 0.8202 0.33 12

4.0 4.1 4.101

12.7. EARLY COMPENSATORS This section on null compensators is adapted from the original chapter in earlier versions of this book, written by late Abbe Offner. The corresponding test of a

3.1 1 PROBLEMS 3.1 Show for discrete stochastic variables that the expectation of the Fisher score vector is equal to the null vector.

print_r( $authors ); echo $nextCell; print_r( array_splice( $authors, 2, 0, $arrayToAdd ) ); echo $nextCell; print_r( $arrayToAdd ); echo $nextCell; print_r( $authors ); echo $rowEnd; echo {$headingStart}2. Replacing two elements with a new element{$headingEnd} ; $authors = array( Steinbeck , Kafka , Tolkien ); $arrayToAdd = array( Bronte ); echo $rowStart; print_r( $authors ); echo $nextCell; print_r( array_splice( $authors, 0, 2, $arrayToAdd ) ); echo $nextCell; print_r( $arrayToAdd ); echo $nextCell; print_r( $authors ); echo $rowEnd; echo {$headingStart}3. Removing the last two elements{$headingEnd} ; $authors = array( Steinbeck , Kafka , Tolkien ); echo $rowStart; print_r( $authors ); echo $nextCell; print_r( array_splice( $authors, 1 ) ); echo $nextCell; echo Nothing ; echo $nextCell; print_r( $authors ); echo $rowEnd; echo {$headingStart}4. Inserting a string instead of an array{$headingEnd} ; $authors = array( Steinbeck , Kafka , Tolkien ); echo $rowStart; print_r( $authors ); echo $nextCell; print_r( array_splice( $authors, 1, 0, Orwell ) ); echo $nextCell; echo Orwell ; echo $nextCell; print_r( $authors ); echo $rowEnd; echo </table> ; > </body> </html>

and with xn = b for n [b 2h, b], we have [via (9.105b)] 1 1 [3f (xn ) 4f (xn 1 ) + f (xn 2 )] + h2 f (3) ( n ). 2h 3 (9.106b) Thus, (9.104) becomes (approximate corrected trapezoidal rule) f (1) (xn ) = f (1) (b) = TC (n) = T (n) h [3f (xn ) 4f (xn 1 ) + f (xn 2 ) + 3f (x0 ) 4f (x1 ) + f (x2 )] 24 (9.107)

But x = 0 as it is an eigenvector of A, and also a = 0. Immediately, = , contradicting our assumption that these eigenvalues are distinct. Thus, y = ax is impossible; that is, we have proved that y is independent of x. Theorem 11.2 leads us to the next theorem. Theorem 11.3: If A Cn n has n distinct eigenvalues, then A has n linearly independent eigenvectors. Proof Uses mathematical induction (e.g., Stewart [6]). We have already seen the following theorem. Theorem 11.4: If A Rn n and A = AT , then all eigenvalues of A are realvalued. Proof See Hill [1], or see the appropriate footnote in 4. In addition to this theorem, we also have the following one. Theorem 11.5: If A Rn n , and if A = AT , then eigenvectors corresponding to distinct eigenvalues of A are orthogonal. Proof Suppose that ( , x) and ( , y) are eigenpairs of A with = . We wish to show that x T y = y T x = 0 (recall De nition 1.6). Now x = Ax = AT x so that y T x = y T AT x = (Ay)T x = y T x, implying that ( )y T x = 0. But = so that y T x = 0, that is, x y. Theorem 11.5 states that eigenspaces corresponding to distinct eigenvalues of a symmetric, real-valued matrix form mutually orthogonal vector subspaces of Rn . Any vector from one eigenspace must therefore be orthogonal to any eigenvector from another eigenspace. If we recall De nition 11.2, it is apparent that all symmetric, real-valued matrices are nondefective, if their eigenvalues are all distinct.1 In fact, even if A Cn n and is not symmetric, then, as long as the eigenvalues are distinct, A will be nondefective (Theorem 11.3).

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