Definite matrix

Creator
Creator
Seonglae Cho
Created
Created
2024 Oct 21 13:35
Editor
Edited
Edited
2024 Nov 18 12:22
Refs
Refs

Quadratic form Property

Important application of positive definiteness is
Convex Optimization
. For a twice differentiable real-valued function f, if the Hessian matrix is a positive definite matrix, then the overall shape of this function is convex downward, and it necessarily has a global minimum. If we calculate the gradient here and find the point where it is zero, that point is precisely the local minimum. This is the multivariate version of the second derivative test for
Convex function
.

Negative definite Matrix

x,xTAx<0,x0\forall x,x^TAx < 0, x\ne 0

Positive Definite matrix

x,xTAx>0,x0\forall x, x^TAx > 0, x\ne 0
Semi-definite implies weak
Convexity
or
Concavity

Positive semi-definite matrix

x,xTAx0\forall x,x^TAx \ge 0

Negative semi-definite matrix

x,xTAx0\forall x,x^TAx \le 0
 
 
 
 
 

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