n9 - (a) A is positive definite if and only if all the...

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Atsushi Inoue ECG 765: Mathematical Methods for Economics Fall 2009 Lecture Notes 9 Eigenvalue Tests of Definiteness We will review another test for definiteness to check the second-order suffi- cient condition for optimization. As shown later in the semester, eigenvalues and eigenvectors are crucial for solving differential equations and understand- ing their solutions. Definition. Let A be a square matrix. If a scalar λ and a nonzero vector x satisfy Ax = λx, x 6 = 0 , then λ is called an eigenvalue of A and x is called an eigenvector of A associ- ated with λ . λ and x are also called characteristic vector and characteristic value , respectively. Theorem. Let A be an n × n symmetric matrix. Then
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Unformatted text preview: (a) A is positive definite if and only if all the eigenvalues of A are (strictly) positive. (b) A is positive semidefinite if and only if all the eigenvalues of A are nonnegative. (c) A is negative definite if and only if all the eigenvalues of A are (strictly) negative. (d) A is negative semidefinite if and only if all the eigenvalues of A are nonpositive. (e) A is indefinite if and only if A has a positive eigenvalue and a negative eigenvalue. Suggested Exercises. Read Section 23.1 of Simon and Blume (1994) and work on Exercises 23.1–23.5 of Simon and Blume (1994). 1...
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This note was uploaded on 10/22/2009 for the course ECG 765 taught by Professor Fackler during the Fall '07 term at N.C. State.

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