By Bacco M., Mocellin V.
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Extra info for A bayesian justification for the linear pooling of opinions
The set e 1,…,e n forms a basis for ℝ n , called the standard basis. If x 1,…,x m is a basis for S then any vector x in S admits a unique representation as a linear combination c 1 x 1+⋯+c m x m . For, if then and since x 1,…,x m are linearly independent, c i =d i for each i. A vector space is said to be finite dimensional if it has a basis consisting of finitely many vectors. The vector space containing only the zero vector is also finite dimensional. We will consider only finite dimensional vector spaces.
What can you say about the rank of A? Let A be an n×n positive definite matrix, n>1, and suppose a ij ≤0 for all i≠j. Let B be the Schur complement of a 11 in A. Show that b ij ≤0 for all i≠j. Let A be an n×n matrix, not necessarily symmetric, and suppose all principal minors of A are positive. Show that any real eigenvalue of A must be positive. Let A, B be n×n positive semidefinite matrices and let C be the matrix with its (i,j)-entry given by c ij =a ij b ij , i,j=1,…,n. Show that C is positive semidefinite.
To prove the uniqueness we must show that if B,C are positive semidefinite matrices satisfying A=B 2=C 2, then B=C. Let D=B−C. By the Spectral Theorem, there exists an orthogonal matrix Q such that Z=QDQ′ is a diagonal matrix. Let E=QBQ′, F=QCQ′ and it will be sufficient to show that E=F. Since Z=E−F is a diagonal matrix, e ij =f ij , i≠j. Also, and therefore, If z ii =0 then e ii =f ii . If z ii ≠0 then e ii +f ii =0. However, since E, F are positive semidefinite, e ii ≥0, f ii ≥0 and it follows that e ii =f ii =0.