Definite, Semi-Definite and Indefinite Matrices - Mathonline PDF

Preview

Summary

lecture notes...

Description

Mathonline LEARN MATHEMATICS

Definite, Semi-Definite and Indefinite Matrices Fold Table of Contents Definite, Semi-Definite and Indefinite Matrices Example 1 Example 2

Definite, Semi-Definite and Indefinite Matrices We are about to look at an important type of matrix in multivariable calculus known as Hessian Mat derivatives test for a real-valued function z = f (x1 , x 2 , . . . , xn ) of n variables with continuous par a = (a 1 , a 2 , . . . , a n ) ∈ D(f ) to determine whether f (a) is a local maximum value, local minimum Before we do this though, we will need to be able to analyze whether a square n × n symmetric ma positive/negative semidefinite. These terms are more properly defined in Linear Algebra and relate now go into the specifics here, however, the definition below will be sufficient for what we need.

∣ a 11 ⎡ a 11 a 12 ⋯ a 1n ⎤ ⎢a ⎥ ∣a a 22 ⋯ a 2n ⎥ 21 ⎢ ∣ 21 n × n D Definition: Let A = ⎢ be an symmetric matrix, and let = i ∣ ⋮ ⋮ ⋱ ⋮ ⎥⎥ ⎢ ⋮ ∣ ⎣ a n1 a n2 ⋯ a nn ⎦ ∣ a i1 a) A is said to be Positive Definite if D i > 0 for i = 1, 2, . . . , n. b) A is said to be Negative Definite if D i < 0 for odd i ∈ {1, 2, . . . , n} and D i > 0 for even i c) A is said to be Indefinite if det(A) = D n ≠ 0 and neither a) nor b) hold. d) If det(A) = D n = 0, then A may be Indefinite or what is known Positive Semidefinite or Neg The values D i for i

= 1, 2, . . . , n are the values of the determinants of the i × i top left submatrice

Let's look at some examples of classifying square symmetric matrices.

Example 1 Classify the following square symmetric matrix We have that D 1

= 6 > 0 and D

∣6 ∣

4∣ ∣

A= 30

6 [4 16

4 5 ] as positive definite, negative d 14 > 0 Therefore A is a positive d...