(ii) (5 %) Show that there exists a matrix, which is similar to the matrixin the Jordon form, i.e., 

7.美國為了給予南海主權之爭的國家戰略保證所採取的「航行自由行動」，可能產生哪些影響？其中，我國可以做些什麼？

8.「中」印邊界衝突擴大可能對我國帶來什麼影響？

Problem 1. (10 %) Given a matrixwhere a is a real number. Find the rank ofand give the corresponding range of a.

(i) (10 %) Find the Eigenvalues and Eigenvectors for .

(i) (15 %) Letwith r singular valuesUr= [u1,⋯,ur]consisting of the associated r left singular vectors and Vr= [v1,⋯,vr]consisting of the associated r right singular vectors. (1) Find the range space (i.e., column space) of A, denoted as R(A), and the row space of A (i.e., R(AT)) where AT denotes the transpose of A ；(2) find the projection matrix PA such thatand；(3) find the eignevalues and eigenvectors of AAT.

(ii) (10 %) Suppose that A,and Tr(A) denotes the trace of A (i.e., the sum of all the diagonal elements of A).(1)Is Tr(AX)≥0 true? (2)Is it true that if Tr(AX) = 0, then(zero matrix)?Prove or disprove your answer.

1.國防政策並非局限於____________，任何與維護國防安全的事物都屬於國防政策範疇。

2.非傳統安全威脅與日俱增，國軍秉持________________________原則，守護家園及人民福祉。

3.為貫徹國防法制化，秉持「法制先行」原則，需研訂__________________及__________________，與司法機關建立現役軍人涉法案件通報機制。

4.____________強調以軍事武力及資源，達成國防政策的布局與安排。