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Unformatted text preview: Notes on linear algebra (Monday 17th October. 2016. 23:10) page 149 identical (since AM 2 A22 2 . . . = Ak +13 +1 2 1). Also, A is a permutation matrix. The proof of Lemma 3.126 (c) is thus complete. (d) Let a and a be two distinct elements of {k -|— 1,]: —|— 2.....11} satisfying a = k + 1 and Am 2 1. We must prove that 1"“va is a (k -|— 1)-identical permutation matrix. Let B = Ta,oA- Proposition 3.105 (applied to m = a and C = A) shows that TWA is the a x a-matrix obtained from A by swapping the a-th row with the o-th row. Since B : TWA, this rewrites as follows: B is the a X a-matrix obtained from A by swapping the a-th row with the v-th row. Hence. Lemma 3.117 shows that B is a permutation matrix. We shall next show that B is (k + 1)-identical. Recall that B is the a x n-matrix obtained from A by swapping the a-th row with the v-th row. Hence, the following facts hold: Fact 4: The u-th row of the matrix B equals the v-th row of A. Fact 5: The v-th row of the matrix B equals the a-th row of A. Fact 6: Ifi 6 {1,2,. . .,1a} is such that 1' 7E a and i 7é “a, then the i-th row of the matrix B equals the i-th row of A. Now, using Fact 6. we can easily see that Bi},- 2 Afff for each i 6 {1,2,. . ., k} 9‘3. Hence. for each i E {1.2 ..... kl. we have B"; = An = 1 (bV (128)). In other words. ...
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