(i) Let f(a) be a polynomial such that f(A) = Onxn. Convince me that m(a) f(a)....

 

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(i) Let f(a) be a polynomial such that f(A) = Onxn. Convince me that m₁(a) f(a). (ii) Up to similarity, classify all 5 x 5 matrices such that A² - 5A = -615. [Hint: Let f(a) = a² - 5a + 6. Hence, by hypothesis, f(A) = 05x5. By (i), mA(a) | f(a). Hence m₁(a) = a - 2 OR mA(a) = a-3 OR mд(a) = f(a). ] To 0 0 1 (iii) Let A be a 4 x 4 such that A is similar to H = 00 1 0 0 0 0 1 9 - 18 8 2 a. Find CA (a) and mA (a). b. For each eigenvalue a of A find dim (Ea (A))[ Hint: C₁(a) = (a² - 2a + 1)(a²-9)] c. Theoretically, how do you construct the columns of the invertible matrix Q where Q-¹AQ = H? (i) Let f(a) be a polynomial such that f(A) = Onxn. Convince me that m₁(a) f(a). (ii) Up to similarity, classify all 5 x 5 matrices such that A² - 5A = -615. [Hint: Let f(a) = a² - 5a + 6. Hence, by hypothesis, f(A) = 05x5. By (i), mA(a) | f(a). Hence m₁(a) = a - 2 OR mA(a) = a-3 OR mд(a) = f(a). ] To 0 0 1 (iii) Let A be a 4 x 4 such that A is similar to H = 00 1 0 0 0 0 1 9 - 18 8 2 a. Find CA (a) and mA (a). b. For each eigenvalue a of A find dim (Ea (A))[ Hint: C₁(a) = (a² - 2a + 1)(a²-9)] c. Theoretically, how do you construct the columns of the invertible matrix Q where Q-¹AQ = H?

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i To prove that maa divides fa where fa is a polynomial and maa is the minimal polynomial of a we can use the polynomial division algorithm Lets assume that fa qamaa ra where qa and ra are polynomials ... View the full answer