【正文】 s called the adjoint matrix of A (see[6,]for ??Aadj ). (v) ijE in ? ?RMn denotes the matrix obtained from the identity matrix IIn? by interchanging row i and row j ? ?ji? . Lemma 1 Let ? ?? ?3?? nRMA n . Then we have that ? ? ? ?? ? ijijijij EAadjEAEE ?a d j . We omit the proof of Lemma 1. Lemma 2 We assume that 3?n . Let ? ? ? ?RMaA nij ?? be a real matrix of order n defined by ?????????????????.,10,2,1an d0,211njinifnjijiifnjiifa ij Assume that A is we have the following: (i) If 0,1 ?? ina for 21 ??? ni , then ? ? .0a n d310,1 ?????? inin ania ， (ii) If ? ?110 ???? nia in , then ? ? .0a n d, 1, ????? ?nijn anja (iii) If ? ?1101, ????? nia ni , then ? ? .0a n d,310 ,1,1 ????? ?? injn anja (iv) If ? ?2101 ???? nia n , then ? ? 0a n d,310 ,1 ????? ? innj anja We omit the proof of Lemma2. Example 1 We list 18 real 55 idempotent matrices A with ? ? 3?Ar 24 (the rank of A is equal to 3). In this example, ????????????????0000000001000001000001baz denotes a 55 real idempotent matrix A with ? ? 5?Ar .0a nd,0,0a nd ??? zba In this matrix, we can add that: (i) If 0,0 ?? ba and all other entries of 54 and AA are zero, then we can have 034 ??za and all other entries of AA 54 and must be zero, where iA and Ai denote respectively the i th row and i th column of A . (ii) Similarly, if 0?z and all other entries of AA 54 and are zero, then only nonzero entries are 5251 an d abaa ?? . (1) (2) (3) ????????????????0000000000100001000001awv ????????????????0000000000100000100001bwu ????????????????0000000000010000100001cvu (4) (5) (6) ????????????????0000000001000001000001baw ????????????????0000000000100001000001cav ????????????????0000000000100000100001cbu (7) (8) (9) 25 ????????????????0000000001000001000001cba ????????????????0000000001000001000001cba ????????????????0000000000100001000001azy (10) (11) (12) ????????????????0000000000100000100001bzx ????????????????0000000000010000100001cyx ????????????????0000000001000001000001baz (13) (14) (15) ????????????????0000000000100001000001cay ????????????????0000000000100000100001cbx ????????????????0000000000010000100001zyx (16) (17) (18) ????????????????0000000000010000100001zyx ????????????????0000001000001000001fedcba ????????????????0000000000100010001wzvyux We quote the following [2]: 26 If A is a real idempotent matrix of order n , then A is similar to a diagonal matrix ? ?00,11,1d ia g ， ?? , that is ? ?? ? 10,0,1,1,1d ia g ?? TTA ?? where T is a nonsingular matrix of order n . We prove the following theorem. Theorem Let ? ? ? ?RMaA nij ?? be a real idempotent matrix. Then the adjoint matrix ??Aadj is idempotent. Proof The proof consists of several steps. (i) Suppose the rank ??Ar of a real idempotent matrix A of order n is equal to n . Then we see that IIA n ?? , the identity matrix. We can pute ? ?Aadj as I and hence ? ? IAadj ? is idempotent. (ii) Suppose that ? ? 1??nAr . Then we can assume, without loss of generality, that ?????????????????? 00100000010000011321 naaaaA?????????? We can pute ??Aadj , the adjoint matrix of A as follows: ? ??????????????????????? 00100000010000011321 naaaaAa d j?????????? 27 We can show that ??Aadj is idempotent. (iii) We referring to Lemma 1 and 2, and Example 1, we claim that if A is an idempotent of rank ? ?2?n , then one of the following three statements holds: (1) A has two rows? ?21 and ?? nn AA each of two is the zero vector. (2) A has two columns ? ?AA nn 21 and ?? each of two is the zero vector. (3) A has one row and one column ? ?AAAA nnnn 11 a n dora n d ?? ， both of them are zero vectors. (iv) We just prove that the claim mentioned in the above (iii) is true (for a case). Suppose thatA is an idempotent of rank of ? ?2?n and suppose, in addition, that 2,2,1,1 ??? nia ii ? for 1an d,0 ??? nia ii and ni? . Then we can show that 3,a n df o r,0 ???? njijia ij , and njinjia ij ????? ,1a n df o r,0 . Now if ? ?210,1 ????? nia in , then An1? (the 1?n column) must be the zero vector. In addition, if ? ?210 ???? nja nj , then An (the n column) must also be the vector. This proves the claim for a case (referring to 17, Example1). The rest of all other cases will be proved by a similar way using Lemma 2. 28 Now we see that ? ? ? ?RMAadj n?? 0 for an idempotent matrix of order 2?n , where 0 denotes the zero matrix. Therefore ??Aadj is idempotent. (v) LetA be an idempotent matrix of r