要問線性代數---正定的問題

Thm:(Sylvester Law of Inertia) if A,X:實數矩陣, A:symmetric, X:nonsingular, then A and X^t*A*X have the same inertia. ------------------------------------------------------------------------------ B positive define and symmetric =>B^(-1) is also positive definite and symmetric =>設B^(-1) = L*D*L^t = K*K^t, where K=L*D^(1/2) is invertible then M = B^(-1)*C = K*K^t*C, and let R:= K^(-1)*M*K = K^(-1)*K*K^t*C*K=K^t*C*K then consider (1)R=K^(-1)*M*K===>M and R have the same eigenvalues and because eig(M)>0 so we have eig(R) 皆大於0 (2)R=K^t*C*K ===> by Sylvester's law of inertia, K:invertible, C:symmetric then R and C have the same inertia. 因為eigenvalues of R 都大於0,所以C的eigenvalues也都大於0了!!