n8 - Atsushi Inoue ECG 765: Mathematical Methods for...

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Unformatted text preview: Atsushi Inoue ECG 765: Mathematical Methods for Economics Fall 2009 Lecture Notes 8 Determinantal Tests of Definiteness To test the second-order sufficient condition we need to find out the definite- ness of the Hessian matrix of an objective function. There are two commonly used methods to test definiteness of a matrix. One is based on determinants and another is on eigenvalues. Outline A. Determinants B. Principal Minors and Definiteness A. Determinants Definition. Let A = " a b c d # be a 2 × 2 matrix. We define its determinant to be ad- bc . The determinant of A is written in two ways, det A and | A | . Definition. Let A = a 11 a 12 a 13 a 21 a 22 a 23 a 31 a 32 a 33 be a 3 × 3 matrix. We define its determinant according to the formula as the expansion by a row, say the first row. That is, we define det A = a 11 a 22 a 23 a 32 a 33- a 12 a 21 a 23 a 31 a 33 + a 13 a 21 a 22 a 31 a 32 = a 11 | A 11 | - a 12 | A 12 | + a 13 | A 13 | ....
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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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n8 - Atsushi Inoue ECG 765: Mathematical Methods for...

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