96下線代期末考 é›&ra

96下線代期末考 é›&ra

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1. Solution: scores (a). True The eigenspace of A corresponding to eigenvalue λ is the null space of AI . Since 22 A × ∈ℜ and with an eigenspace of dimension 2, rank () 0 −= , or, equivalently, ( ) 0 = . 5 (a). False D is a diagonal matrix and its diagonal entry ii d is the eigenvalue of A corresponding to column i p , eigenvector corresponding to ii d . If we change the permutation of the columns of P , the order of the diagonal entries of D will change. Ex. 11 13 231023 322032 02 010201 100110 A ⎡⎤ == = ⎢⎥ ⎣⎦ 5 2. Solution: scores Assume that n x ^ satisfies 0 Ax = , then ( ) 0 TT AA x AA x = = . Thus () ( ) Null Null T A (1) Assume that n x ^ satisfies 0 T AAx = , then ( ) 2 0 T T x A Ax x A Ax Ax = 0 = . Thus ( ) Null Null T A (2) From (1) and (2), we have Null Null T A = Nullity of Nullity of T A = () ( ) ( ) rank rank Nullity of T A AA n A (3) 10
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3. Solution: scores (a) The ii element of AB is 1 n ij ji j ab = , and the element of BA is 1 n ij ji j ba = . Thus () 11 trace trace nn ij ji ji ij ij ji ij AB a b b a b a BA == ==== ∑∑ . 6 (b) Let B × ^ such that 1 CB D B = (1) where 1 2 1 00 0 0 n n D λ ⎡⎤ ⎢⎥ = ⎣⎦ " %% #%%% # " (2) () ( ) ( ) ( ) ( ) () () 12 det det det det det det det det det n D B B D B B DD B λλ −− = " (3) 3 1 trace trace trace trace n i i D B B B = === = (4) 3 ( ) ( ) ( ) 1 1 1 trace trace trace trace trace trace k k k n kk k k i i D B B D B B D B B D B BD B B BD D
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96下線代期末考 é›&ra

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