By Picard E., Simart G.

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3) Let A be an n-by-n diagonal matrix whose entries are not necessarily distinct. Suppose A has a diagonal entry d1 that appears k1 times, a diagonal entry d2 that appears k2 times, . , and a diagonal entry dm that appears km times. Then, exactly by the argument of (2), we have that mA (x) = (x − d1 ) · · · (x − dm ) and cA (x) = (x − d1 )k1 · · · (x − dm )km . (4) Let J be an n-by-n Jordan block with diagonal entry a. Let E = {e1 , . . , en } be the standard basis of Cn . Then (J − aI )e1 = 0 and (J − aI )ei = ei−1 for i > 1.

28 CHAPTER 2. THE STRUCTURE OF A LINEAR TRANSFORMATION The coefficient of w1 in this expression is r1,t + ct λ1 − ct rt,t = r1,t + ct (λ1 − rt,t ). Note that λ1 − rt,t = 0, as the first k = k1 diagonal entries of R are equal to λ1 , but the remaining diagonal entries of R are unequal to λ1 . Hence if we choose ct = −r1,t /(λ1 − rt,t ) we see that the w1 -coefficient of T (ut ) is equal to 0. In other words, the matrix of T in the basis {w1 , . . , wk , uk+1 , wk+2 , . . , wn } is of the same form as R, except that the entry in the (1, k + 1) position is 0.

Rk,t wk + rt,t (ut − ct w1 ) = (r1,t + ct λ1 − ct rt,t )w1 + r2,t w2 + . . rk,t wk + rt,t ut . 28 CHAPTER 2. THE STRUCTURE OF A LINEAR TRANSFORMATION The coefficient of w1 in this expression is r1,t + ct λ1 − ct rt,t = r1,t + ct (λ1 − rt,t ). Note that λ1 − rt,t = 0, as the first k = k1 diagonal entries of R are equal to λ1 , but the remaining diagonal entries of R are unequal to λ1 . Hence if we choose ct = −r1,t /(λ1 − rt,t ) we see that the w1 -coefficient of T (ut ) is equal to 0. In other words, the matrix of T in the basis {w1 , .

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