Hey there! Sign in to join this conversationNew here? Join for free
x Turn on thread page Beta
    • Thread Starter
    Offline

    1
    ReputationRep:
    Hi,

    Having trouble wrapping my head around working out whether a hessian matrix is positive, positive semi, negative or negative semi definite.

    E.g. a 2x2 matrix:

    a11 a12
    a21 a22

    am I right in thinking it

    is positive definite if a11 > 0 and the determinant > 0 (minimum)
    is negative definite if a11 < 0 and determinant is > 0 (maximum)
    indefinite if the determinant is < 0 (saddle point)

    Have no idea how to check for semi definite though, if anyone could help or correct me it would be much appreciated
    • PS Helper
    • Study Helper
    Offline

    13
    ReputationRep:
    PS Helper
    Study Helper
    (Original post by Scorcher)
    Hi,

    Having trouble wrapping my head around working out whether a hessian matrix is positive, positive semi, negative or negative semi definite.

    E.g. a 2x2 matrix:

    a11 a12
    a21 a22

    am I right in thinking it

    is positive definite if a11 > 0 and the determinant > 0 (minimum)
    is negative definite if a11 < 0 and determinant is > 0 (maximum)
    indefinite if the determinant is < 0 (saddle point)

    Have no idea how to check for semi definite though, if anyone could help or correct me it would be much appreciated
    No. The matrix {{5, 17}, {0, 11}} is indefinite: b = (1, -\frac{17}{22} + \frac{\sqrt{69}}{22}) gives b^T M b = 0, but det M is 55.

    Possibly the quickest way to work out these things: a real matrix is positive/negative definite iff the symmetric matrix \frac{1}{2} (M+M^T) is pos/neg def. A symmetric matrix is pos/neg def iff all its eigenvalues are positive/negative.
 
 
 
Reply
Submit reply
Turn on thread page Beta
Updated: December 24, 2014
Poll
Are you going to a festival?
Useful resources

Make your revision easier

Maths

Maths Forum posting guidelines

Not sure where to post? Read the updated guidelines here

Equations

How to use LaTex

Writing equations the easy way

Student revising

Study habits of A* students

Top tips from students who have already aced their exams

Study Planner

Create your own Study Planner

Never miss a deadline again

Polling station sign

Thinking about a maths degree?

Chat with other maths applicants

Can you help? Study help unanswered threads

Groups associated with this forum:

View associated groups

The Student Room, Get Revising and Marked by Teachers are trading names of The Student Room Group Ltd.

Register Number: 04666380 (England and Wales), VAT No. 806 8067 22 Registered Office: International House, Queens Road, Brighton, BN1 3XE

Write a reply...
Reply
Hide
Reputation gems: You get these gems as you gain rep from other members for making good contributions and giving helpful advice.