Given a matrix a=(a1)nxn,fori1,…,i∈{1,2,…,n} with i1<i2< define a kx k minor as In particular, denote the principal minors as di≡d(},d≡d(12},……,dn=d(12,,n Theorem 2.1. For a symmetric matrix A, (a)A>0←→k>0, for all k. (b)A<0←→(-1)dk>0, for all k (c)A≥0÷→di}≥0, for all permutations{in,,ik} and all k=1, (d)A≤0←→(-1)dxa}≥0, for all permutations{iy…,ik} and all k=1,…,n Example 2. (a)Verify >0. b (b) Find conditions for A Proposition 2.1. a>0 iff all its eigenvalues are positive a>0 iff all its eigenvalues are strictly positive If A>0 then A-I>0 A>0今-A<0Given a matrix A = (aij )n×n, for i1, ··· , ik ∈ {1, 2, ··· , n} with i1 < i2 < ··· < ik, define a k × k minor as d{i1,···ik} ≡                ai1i1 ai1i2 ··· ai1ik ai2i1 ai2i2 ai2ik . . . . . . . . . aiki1 aiki1 ··· aikik                . In particular, denote the principal minors as d1 ≡ d{1}, d2 ≡ d{1,2}, ..., dn ≡ d{1,2,...,n}. Theorem 2.1. For a symmetric matrix A, (a) A > 0 ⇐⇒ dk > 0, for all k. (b) A < 0 ⇐⇒ (−1)kdk > 0, for all k. (c) A ≥ 0 ⇐⇒ d{i1,···ik} ≥ 0, for all permutations {i1,...,ik} and all k = 1, . . . , n. (d) A ≤ 0 ⇐⇒ (−1)kd{i1,···ik} ≥ 0, for all permutations {i1,...,ik} and all k = 1, . . . , n.  Example 2.4. (a) Verify ⎛ ⎜⎝ 1 −1 −1 1 ⎞ ⎟⎠ ≥ 0. (b) Find conditions for A = ⎛ ⎜⎝ a b b c ⎞ ⎟⎠ ≥ 0.  Proposition 2.1. • A ≥ 0 iff all its eigenvalues are positive. • A > 0 iff all its eigenvalues are strictly positive. • If A > 0, then A−1 > 0. • A ≥ 0 ⇔ −A ≤ 0. 2—2