- #1
anbhadane
- 13
- 1
In Jackson,(3rd edition) Chapter 1 , page no, 44 He uses the word "semipositive definite" what is it? is it "non-negative" definite?
Semipositive definite is a mathematical term used to describe a matrix that has certain properties. Specifically, it refers to a square matrix where all of the eigenvalues are greater than or equal to zero.
Eigenvalues are a set of numbers associated with a matrix that represent the scale factor by which a corresponding eigenvector is stretched or compressed. In other words, they are the solutions to the characteristic equation of a matrix.
While both positive definite and semipositive definite matrices have all positive eigenvalues, the main difference is that positive definite matrices have all positive eigenvalues and semipositive definite matrices have all non-negative eigenvalues. This means that a semipositive definite matrix can have some zero eigenvalues, while a positive definite matrix cannot.
Semipositive definite matrices have a variety of applications in fields such as physics, engineering, and statistics. They are commonly used in optimization problems, signal processing, and in the analysis of systems with multiple variables.
To determine if a matrix is semipositive definite, you can use the Cholesky decomposition or the Sylvester's criterion. The Cholesky decomposition involves factoring the matrix into a lower triangular matrix and its conjugate transpose, while Sylvester's criterion states that a matrix is semipositive definite if and only if all of its leading principal minors are non-negative.