Positive operator?

In summary, a positive operator is a type of mathematical operator that preserves the positivity of vectors. It is different from a non-negative operator in that it only allows for strictly positive values. Some examples of positive operators include the identity and transpose operators, and they have applications in various fields of science, such as quantum mechanics and signal processing. In quantum mechanics, they are used to represent observables and density matrices, and to define the dynamics of a quantum system.
  • #1
Seckin Sefi
6
0
"A positive operator A is defined to be an operator such that for any vector |v>=!0, <v|A|v> is real, non-negative number."

Can somebody tell me how can I check if a matrice (for example 4x4) is positive or not?

Thanks in advance
 
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  • #2
This behaviour also comes under "positive definite", ie A defines a positive definite bilinear map. Google along those lines, it's well documented,

One criterion is if the eigenvalues are all positive.
 
  • #3
for your help!

Sure, to check if a matrix is positive, you can use the definition of a positive operator stated above. First, you would need to find the eigenvalues of the matrix. If all the eigenvalues are real and non-negative, then the matrix is positive. If any of the eigenvalues are negative, then the matrix is not positive. Another way to check is by using the Sylvester's criterion, which states that a matrix is positive if and only if all the leading principal minors (determinants of the top-left submatrices) are positive. I hope this helps!
 

1. What is a positive operator?

A positive operator is a mathematical concept used in linear algebra and functional analysis. It is an operator that maps a vector space to itself and preserves the positivity of the vectors, meaning that it maps positive vectors to positive vectors.

2. How is a positive operator different from a non-negative operator?

A non-negative operator allows for the possibility of zero values, while a positive operator only allows for strictly positive values. In other words, a non-negative operator can map a vector to a vector with some zero components, while a positive operator can only map a vector to a vector with all positive components.

3. What are some examples of positive operators?

Some examples of positive operators include the identity operator, which maps a vector to itself, and the transpose operator, which maps a vector to its transpose. Other examples include the Laplace operator, the gradient operator, and the Fourier transform operator.

4. What is the significance of positive operators in science?

Positive operators have many applications in science, particularly in quantum mechanics and signal processing. They are used to represent physical quantities such as energy, momentum, and probability, and play a crucial role in the study of quantum systems and the analysis of signals.

5. How are positive operators used in quantum mechanics?

In quantum mechanics, positive operators are used to represent observables, which are physical quantities that can be measured. They are also used to represent density matrices, which describe the state of a quantum system, and to define the dynamics of a quantum system through the Schrödinger equation.

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