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# 11 consider an n k matrix a a a 11 a 12 a 1 k a 21 a

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11. Consider an n × k matrix A : A = a 11 a 12 · · · a 1 k a 21 a 22 · · · a 2 k a 31 · · · a 3 k . . . . . . . . . . . . a n 1 a n 2 · · · a nk a ) Show that its own inner product A 0 A is symmetric. Show that their diagonal elements are given by A 0 A ii = n X j =1 a 2 ji where A 0 A ii denotes the i th diagonal element of A 0 A . b ) Show that its outer product AA 0 is also symmetric. Find their diagonal elements, and show that they can not be negative. 12. Let A and B be n × k and k × n matrices respectively. Show that tr( AB ) = tr( BA ). 13. Given the same set of matrices as in Q6, calculate the following traces of products. a ) tr( BC ) b ) tr( C 0 B 0 ) c ) tr( CB ) d ) tr[( BC ) 0 ] e ) tr( B 0 C 0 ) 2
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