How Can I Determine if a Matrix is Positive?

[Seckin Sefi]

"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

 

[matt grime]

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.

 

[M98Ranger]

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!