Positive definite function,semi-definite functions

In summary, 'positive definite function' and 'semi-definite function' are terms used in stability analysis of non-linear models. A Lyapunov function V is positive definite and its derivative dV/dt can be either negative or positive semi-definite. These terms refer to the behavior of the function and its derivative, and there is a connection between 'positive definite function' and 'positive definite matrix'.
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marellasunny
255
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Hello all!
*please explain the terms 'positive definite function' and 'semi-definite function'.*
CONTEXT:
I am reading a book on the stability analysis of non-linear models.
In the chapter for 'neighborhood stability analysis',I came across the "Lyapunov function V".

V has the following properties:
1.V is positive definite.
2.dV/dt is negative semi-definite(stable valley)
3.dV/dt is positive semi-definite(unstable valley)

I understand the usual hilltop valley visualization,but please explain the terms 'positive definite function' and 'semi-definite function'. Any level of math is understandable.
**Is there a connect between 'positive definite function' and 'positive definite matrix'?**
 
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What is a positive definite function?

A positive definite function is a mathematical function that satisfies the condition that for any input, the output is always a positive number. In other words, the function returns values that are greater than zero for all possible inputs.

What is a semi-definite function?

A semi-definite function, also known as a positive semi-definite function, is a mathematical function that satisfies the condition that for any input, the output is either positive or zero. In other words, the function returns values that are greater than or equal to zero for all possible inputs.

What are the applications of positive definite and semi-definite functions?

Positive definite and semi-definite functions have various applications in mathematics, physics, and engineering. They are commonly used in optimization problems, signal processing, and quantum mechanics.

What is the difference between positive definite and semi-definite functions?

The main difference between positive definite and semi-definite functions is that positive definite functions always return positive values, while semi-definite functions can return positive or zero values. Additionally, positive definite functions are often used to prove the convergence of series and integrals, while semi-definite functions are used in optimization problems.

How do you determine if a function is positive definite or semi-definite?

To determine if a function is positive definite or semi-definite, we need to look at the eigenvalues of the function's corresponding matrix. If all eigenvalues are positive, the function is positive definite. If all eigenvalues are non-negative, the function is semi-definite. If any eigenvalue is negative, the function is not positive definite or semi-definite.

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