# Linear Algebra question.

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:

## Answers and Replies

Gokul43201
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.

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.

Gokul43201
Staff Emeritus
Science Advisor
Gold Member
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* = ?