Linear Algebra question.

  • #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:

Answers and Replies

  • #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.
 
  • #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.
 
  • #4
Gokul43201
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loop quantum gravity said:
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.
So, if P=SS*, what is P* = ?
 

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