= = Let G be an n x n symmetric matrix with the spectral decomposition G=...

 

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= = Let G be an n x n symmetric matrix with the spectral decomposition G= QTDQ, where Q is an n x n orthogonal matrix (i.e. QQQQ I), D= diag(A... An) is an n xn diagonal matrix whose diagonal elements, A... An are the eigenvalues of the matrix G, and I is an n x n identity matrix. i) Show that if G is positive semi-definite then A. > 0, for i = 1, 2,..., n. ii) Show that if A, > 0, for i = 1, 2,..., n, then G is positive definite. = = Let G be an n x n symmetric matrix with the spectral decomposition G= QTDQ, where Q is an n x n orthogonal matrix (i.e. QQQQ I), D= diag(A... An) is an n xn diagonal matrix whose diagonal elements, A... An are the eigenvalues of the matrix G, and I is an n x n identity matrix. i) Show that if G is positive semi-definite then A. > 0, for i = 1, 2,..., n. ii) Show that if A, > 0, for i = 1, 2,..., n, then G is positive definite.