How to make matrix positive definite (when it is not)?

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This discussion focuses on methods to transform a non-positive definite matrix into a positive definite one. The primary approach involves obtaining the eigen decomposition of the matrix and replacing any negative eigenvalues with zeros. Singular value decomposition was also mentioned as a potential method, but the consensus is that adjusting eigenvalues is the most direct solution. The discussion concludes that if the matrix is not positive definite, it is essential to understand the relationship between the original and modified matrices.

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  • Understanding of eigen decomposition
  • Familiarity with matrix properties, specifically positive definiteness
  • Knowledge of eigenvalues and their significance in matrix analysis
  • Basic concepts of singular value decomposition (SVD)
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This discussion is beneficial for mathematicians, data scientists, and engineers working with matrix computations, particularly those involved in optimization and numerical analysis.

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Suppose I have a matrix that looks like this
[,1] [,2]
[1,] 2.415212e-09 9.748863e-10
[2,] -2.415212e-09 5.029136e-10

How do I make it positive definite? I am not looking for specific numerical value answer, but a general approach to this problem.

I have heard singular value decomposition, or getting some eigenvalue? Is that correct?
 
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What do you mean by "make" a matrix positive definite? Since this matrix is NOT positive definite you must mean to change it into one that is. What is the relationship of this new matrix to the original supposed to be?
 
HallsofIvy said:
What do you mean by "make" a matrix positive definite? Since this matrix is NOT positive definite you must mean to change it into one that is. What is the relationship of this new matrix to the original supposed to be?

I was hoping to "make" it positive using some trick, but after looking around again I am wrong. Like you said, if it's NOT positive definite, then it's not.

I also realized there's an error when putting together the matrix. So, problem solved I guess.
 
From an engineer point of view what I would do if I had a non-positive definite matrix is:
  1. Obtain its eigen decomposition.
  2. Changes the negative eigenvalues for zeroes.
 

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