Let a be a real symmetric idempotent matrix of order n, that is, a & # 178; = a (1) It is proved that there is an orthogonal matrix Q such that (Q-1) AQ = diag (1,1,...) ,1,0,…… ,0） (2) If the rank of a is r, DET (a-2i) is calculated

(1) A is a real symmetric idempotent matrix of order n, so the eigenvalues of a can only be 0 and 1
So there is an orthogonal matrix Q such that (Q-1) AQ = diag (1,1,...) ,1,0,…… ,0）
(2) Let 1 be R-fold and 0 be n-r-fold,
Then the matrix a-2i has R multiple eigenvalues 1-2 = - 1 and N-R multiple eigenvalues 0-2 = - 2
So det (a-2i) = (- 1) ^ n * 2 ^ (N-R)

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