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Homework Help: Linear Algebra question.

  1. Jun 6, 2006 #1

    MathematicalPhysicist

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    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

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    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

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    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

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    So, if P=SS*, what is P* = ?
     
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