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# Week_12 - Week 12 Section 5.5 Similarity&...

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Unformatted text preview: Week 12 Section 5.5: Similarity & Diagonalization Revisited Definition: If A and B are n × n matrices, then we say A and B are similar , denoted A B if B = P- 1 AP . Note: A is diagonalizable iff A is similar to a diagonal matrix D since D = P- 1 AP . Definition: If A is an n × n matrix, then the trace of A denoted tr( A ) is the sum of the entries along the main diagonal of A . ie. If a ij denotes the ( i, j )-entry of A , then tr( A ) = a 11 + a 22 + . . . + a nn . Lemma: If A and B are n × n matrices, then tr( AB ) = tr ( BA ). Proof: Let a ij and b ij denote the ( i, j )-entries of A and B , respectively. tr( AB ) = a 11 b 11 + a 12 b 21 + . . . + a 1 n b n 1 + a 21 b 12 + a 22 b 22 + . . . + a 2 n b n 2 + . . . + a n 1 b 1 n + a n 2 b 2 n + . . . + a nn b nn tr( BA ) = b 11 a 11 + b 12 a 21 + . . . + b 1 n a n 1 + b 21 a 12 + b 22 a 22 + . . . + b 2 n a n 2 + a n 1 a 1 n + b n 2 a 2 n + . . . + b nn a nn Comparing the terms in the first column sum of tr( AB ) and the first row sum of tr( BA ) we have the same entries. Similarly, the second column sum of tr( AB ) is equal to the second row sum of tr( BA ) and so on. Lemma: If A is an n × n matrix and B and C are invertible n × n matrix, then Col( AB ) =Col( A ) and Row( CA ) =Row( A ) and thus Rank( AB ) =Rank( A ) =Rank( CA ). Proof: Denote the columns of A by ~ c 1 ,~ c 2 , . . . ,~ c n and let vecx be the j th column of B . Then column j of AB is A~x = x 1 ~ c 1 + x 2 ~ c 2 + . . . + x n ~ c n ∈ Col( A ). Therefore Col( AB ) ⊆ Col( A ). Also, Col( A ) = Col( ABB- 1 ) = Col(( AB ) B- 1 ) ⊆ Col( AB ). 1 Thus Col( A ) = Col( AB ). Row( CA ) = Col(( CA ) T ) = Col( A T C T ) ⊆ Col( A T ) = Row( A ). Also, Row( A ) = Row( C- 1 CA ) = Row( C- 1 ( CA )) = Row( CA ). Thus, Row( CA ) = Row( A ). This tells us that Rank( AB ) = Rank( A ) and Rank( CA ) = Rank( A ). Theorem If A and B are similar, then A and B have the same determinant, rank, trace, characteristic polyno- mial and eigenvalues....
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Week_12 - Week 12 Section 5.5 Similarity&...

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