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Example 2.21.Consider the matricesA=4113,B=⎛⎝111111110.1⎞⎠.The matrixAis positive definite sinceTr(A) =4+3=7>0,det(A) =4·3−1·1=11>0.Downloaded 01/06/17 to 188.8.131.52. Redistribution subject to SIAM license or copyright; see
2.2. Classification of Matrices21As for the matrixB, we haveTr(B) =1+1+0.1=2.1>0,det(B) =0.However, despite the fact that the trace and the determinant ofBare nonnegative, wecannot conclude that the matrix is positive semidefinite since Proposition 2.20 is validonly for 2×2 matrices. In this specific example we can show (even without computingthe eigenvalues) thatBis indefinite. Indeed,eT1Be1>0,(e2−e3)TB(e2−e3) =−0.9<0.For any positive semidefinite matrixA, we can define the square root matrixA12in thefollowing way. LetA=UDUTbe the spectral decomposition ofA; that is,Uis an orthog-onal matrix, andD=diag(d1,d2,...,dn)is a diagonal matrix whose diagonal elements arethe eigenvalues ofA. SinceAis positive semidefinite, we have thatd1,d2,...,dn≥0, andwe can defineA12=UEUT,whereE=diag(d1,d2,...,dn). Obviously,A12A12=UEUTUEUT=UEEUT=UDU=A.The matrixA12is also calledthe positive semidefinite square root.A well-known testfor positive definiteness is theprincipal minors criterion.Given ann×nmatrix, thedeterminant of the upper leftk×ksubmatrix is calledthe kth principal minorand isdenoted byDk(A). For example, the principal minors of the 3×3 matrixA=⎛⎝a11a12a13a21a22a23a31a32a33⎞⎠areD1(A) =a11,D2(A) =deta11a12a21a22,D3(A) =det⎛⎝a11a12a13a21a22a23a31a32a33⎞⎠.The principal minors criterion states that a symmetric matrix is positive definite if andonly if all its principal minors are positive.Theorem 2.22 (principal minors criterion).LetAbe an n×n symmetric matrix. ThenAis positive definite if and only if D1(A)>0,D2(A)>0,...,Dn(A)>0.Note that the principal minors criterion is a tool for detecting positive definiteness ofa matrix. It cannot be used in order detect positivesemidefiniteness.Example 2.23.LetA=⎛⎝423232324⎞⎠,B=⎛⎝22222222−1⎞⎠,C=⎛⎝−4111−4111−4⎞⎠.Downloaded 01/06/17 to 184.108.40.206. Redistribution subject to SIAM license or copyright; see
22Chapter 2. Optimality Conditions for Unconstrained OptimizationThe matrixAis positive definite sinceD1(A) =4>0,D2(A) =det4223=8>0,D3(A) =det⎛⎝423232324⎞⎠=13>0.The principal minors ofBare nonnegative:D1(B) =2,D2(B) =D3(B) =0; however,since they are not positive, the principal minors criterion does not provide any informa-tion on the sign of the matrix other than the fact that it is not positive definite. Since thematrix has both positive and negative diagonal elements, it is in fact indefinite (see Lemma2.16). As for the matrixC, we will show that it is negative definite. For that, we will usethe principal minors criterion to prove that−C