Example 221 consider the matrices a 4 1 1 3 b 1 1 1 1

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Example 2.21. Consider the matrices A = 4 1 1 3 , B = 1 1 1 1 1 1 1 1 0.1 . The matrix A is positive definite since Tr ( A ) = 4 + 3 = 7 > 0, det ( A ) = 4 · 3 1 · 1 = 11 > 0. Downloaded 01/06/17 to 69.91.157.97. Redistribution subject to SIAM license or copyright; see
2.2. Classification of Matrices 21 As for the matrix B , we have Tr ( B ) = 1 + 1 + 0.1 = 2.1 > 0, det ( B ) = 0. However, despite the fact that the trace and the determinant of B are nonnegative, we cannot conclude that the matrix is positive semidefinite since Proposition 2.20 is valid only for 2 × 2 matrices. In this specific example we can show (even without computing the eigenvalues) that B is indefinite. Indeed, e T 1 Be 1 > 0, ( e 2 e 3 ) T B ( e 2 e 3 ) = 0.9 < 0. For any positive semidefinite matrix A , we can define the square root matrix A 1 2 in the following way. Let A = UDU T be the spectral decomposition of A ; that is, U is an orthog- onal matrix, and D = diag ( d 1 , d 2 ,..., d n ) is a diagonal matrix whose diagonal elements are the eigenvalues of A . Since A is positive semidefinite, we have that d 1 , d 2 ,..., d n 0, and we can define A 1 2 = UEU T , where E = diag ( d 1 , d 2 ,..., d n ) . Obviously, A 1 2 A 1 2 = UEU T UEU T = UEEU T = UDU = A . The matrix A 1 2 is also called the positive semidefinite square root . A well-known test for positive definiteness is the principal minors criterion . Given an n × n matrix, the determinant of the upper left k × k submatrix is called the kth principal minor and is denoted by D k ( A ) . For example, the principal minors of the 3 × 3 matrix A = a 11 a 12 a 13 a 21 a 22 a 23 a 31 a 32 a 33 are D 1 ( A ) = a 11 , D 2 ( A ) = det a 11 a 12 a 21 a 22 , D 3 ( A ) = det a 11 a 12 a 13 a 21 a 22 a 23 a 31 a 32 a 33 . The principal minors criterion states that a symmetric matrix is positive definite if and only if all its principal minors are positive. Theorem 2.22 (principal minors criterion). Let A be an n × n symmetric matrix. Then A is positive definite if and only if D 1 ( A ) > 0, D 2 ( A ) > 0,..., D n ( A ) > 0 . Note that the principal minors criterion is a tool for detecting positive definiteness of a matrix. It cannot be used in order detect positive semi definiteness. Example 2.23. Let A = 4 2 3 2 3 2 3 2 4 , B = 2 2 2 2 2 2 2 2 1 , C = 4 1 1 1 4 1 1 1 4 . Downloaded 01/06/17 to 69.91.157.97. Redistribution subject to SIAM license or copyright; see
22 Chapter 2. Optimality Conditions for Unconstrained Optimization The matrix A is positive definite since D 1 ( A ) = 4 > 0, D 2 ( A ) = det 4 2 2 3 = 8 > 0, D 3 ( A ) = det 4 2 3 2 3 2 3 2 4 = 13 > 0. The principal minors of B are nonnegative: D 1 ( B ) = 2, D 2 ( B ) = D 3 ( 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 the matrix has both positive and negative diagonal elements, it is in fact indefinite (see Lemma 2.16). As for the matrix C , we will show that it is negative definite. For that, we will use the principal minors criterion to prove that C

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