I am trying to prove that the following matrix is positive semidefinite: A = [2 1 2; 1 1 1; 2 1 2]. I found that the eigenvalues are 0, 0, and 5. After finding the row echelon form, the pivots are greater than 0. So, this should mean that the matrix is positive semidefinite. But when I try to test the principal submatrices, some of them are less than 0. That means that the matrix is not positive semidefinite.  Am I doing this wrong? Can you prove this is a positive semidefinte matrix with the use of the "No principal submatrices have negative determinants" rule?

Algebra and Trigonometry (6th Edition)
6th Edition
ISBN:9780134463216
Author:Robert F. Blitzer
Publisher:Robert F. Blitzer
ChapterP: Prerequisites: Fundamental Concepts Of Algebra
Section: Chapter Questions
Problem 1MCCP: In Exercises 1-25, simplify the given expression or perform the indicated operation (and simplify,...
icon
Related questions
Question

I am trying to prove that the following matrix is positive semidefinite: A = [2 1 2; 1 1 1; 2 1 2]. I found that the eigenvalues are 0, 0, and 5. After finding the row echelon form, the pivots are greater than 0. So, this should mean that the matrix is positive semidefinite. But when I try to test the principal submatrices, some of them are less than 0. That means that the matrix is not positive semidefinite. 

Am I doing this wrong? Can you prove this is a positive semidefinte matrix with the use of the "No principal submatrices have negative determinants" rule? 

Expert Solution
steps

Step by step

Solved in 2 steps

Blurred answer
Similar questions
Recommended textbooks for you
Algebra and Trigonometry (6th Edition)
Algebra and Trigonometry (6th Edition)
Algebra
ISBN:
9780134463216
Author:
Robert F. Blitzer
Publisher:
PEARSON
Contemporary Abstract Algebra
Contemporary Abstract Algebra
Algebra
ISBN:
9781305657960
Author:
Joseph Gallian
Publisher:
Cengage Learning
Linear Algebra: A Modern Introduction
Linear Algebra: A Modern Introduction
Algebra
ISBN:
9781285463247
Author:
David Poole
Publisher:
Cengage Learning
Algebra And Trigonometry (11th Edition)
Algebra And Trigonometry (11th Edition)
Algebra
ISBN:
9780135163078
Author:
Michael Sullivan
Publisher:
PEARSON
Introduction to Linear Algebra, Fifth Edition
Introduction to Linear Algebra, Fifth Edition
Algebra
ISBN:
9780980232776
Author:
Gilbert Strang
Publisher:
Wellesley-Cambridge Press
College Algebra (Collegiate Math)
College Algebra (Collegiate Math)
Algebra
ISBN:
9780077836344
Author:
Julie Miller, Donna Gerken
Publisher:
McGraw-Hill Education