無年度 - 主題課程_對角化：對角化#107843

科目：主題課程專用 | 選擇題數：0 | 申論題數：13

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申論題 (13)

(a) (3%) Find the eigenvalues of A. ul

(b) (6%) Is A diagonalizable? Justify your answer.

107台大資工

(10%) Your answer will be considered correct only if all the true statements are selected. Let A be an nxn matrix. (a) If x₁ and x₂are the cigenvectors of A, then x₁ + x₂ is also an eigenvector of A.

(b) If A^T = -A, then A is singular.

(c) If A² = A. (A+1)ⁿ = 1 + (2ⁿ+1)A.
(d) If A = A^T, then A is diagonalizable.

and B and D are invertible,then

110交大資訊聯招

b.(10 points) Given a sequence 7, -6, 20, -24, 64, -96,... comes from

for k ≥ 0, with G₀ = 7, G₁= -6 and G₂ = 20. Please find the number

(a) (4%) What are the values of G₄ and G₅?

(b) (5%) Give a matrix A such that

(c) (5%) Find an explicit formula for G_k.

(d) (6 %) What is the limit of G_k as k →∞?

109北科大電子_工數

ㆍ(12%) Find an explicit formula for the sequences defined recursively by

105交大資訊聯招

(6points) Three persons, TC, CC, and YJ, have their own gold. They always divide their own two equal parts and donate to others every day. IC, for example, has 12-kilograms gold today and donate 6 kilograms to Cc and the other 6 kilograms to Y.. If TC has 6 kilograms, CC has 1 kilogram, and YJ has 2 kilograms at first, how many kilograms of gold do CC and YJ have after 365 days?

106交大資訊聯招

(10 points) Political affiliations change from one generation to the next. We consider three parties, labeled D, K, and T. In one generation, membership in D goes 80% to D, 10% to K, and 10% to T； membership in K goes 20% to D, 80% to K, and none to T； and membership in T goes 50% to D, 10% to K, and 40% to T. What is the long-term steady state for the three parties (membership ratios among these three parties)?

(a) (12%) Write down the transition matrix A and calculate its eigenvaluesλ.

(b) (6%) Assume that the initial market share vector of A, B, and C is，what is the limit as k → ∞?