1. Limited time only! Sign up for a free 30min personal tutor trial with Chegg Tutors
    Dismiss Notice
Dismiss Notice
Join Physics Forums Today!
The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

Homework Help: Linear Algebra question.

  1. Jun 6, 2006 #1

    MathematicalPhysicist

    User Avatar
    Gold Member

    prove that a 2x2 complex matrix :
    a b
    c d

    is positive iff A=A*, where A* is the conjugate transpose.

    i know that an operator (and so i think it also applies to a matrix) is positive when it equals SS* for an operator (matrix) S.

    and A* equals:
    \\_
    a c
    _
    b d

    where the upper line stands for the conjugate.
    but i don't know how to find an operator (matrix) which multiplied by its transpose conjugate equals A.
    any pointers will be appreciated.
     
    Last edited: Jun 6, 2006
  2. jcsd
  3. Jun 6, 2006 #2

    Gokul43201

    User Avatar
    Staff Emeritus
    Science Advisor
    Gold Member

    What does it mean to say a matrix is positive? Do you mean positive definite? Are the two terms interchangeable?

    Also, there's another definition that requires that all square sub-matrices have positive determinants - I don't know what this is called.
     
  4. Jun 6, 2006 #3

    MathematicalPhysicist

    User Avatar
    Gold Member

    i know the definition of positive operator (which you can reflect on a matrix cause evey operator can be repesented by a matrix), the definitions is as follows:
    an operator P is positive if it can be represneted by this equation P=SS* where S is an operator.
    the defintion of definite positive requires that S will be non singular.
     
  5. Jun 8, 2006 #4

    Gokul43201

    User Avatar
    Staff Emeritus
    Science Advisor
    Gold Member

    So, if P=SS*, what is P* = ?
     
Share this great discussion with others via Reddit, Google+, Twitter, or Facebook




Loading...
Similar Threads for Linear Algebra question Date
Subspace question May 9, 2016
Linear Algebra -- Projection matrix question Apr 25, 2016
Linear Algebra Question Sep 11, 2015
Linear Transformations, Linear Algebra Question May 10, 2015
Simple matrix/linear algebra question, help Apr 28, 2015