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

[colstat]

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?

 

[HallsofIvy]

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?

 

[colstat]

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.

 

[FoldHellmuth]

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.

 

[blue_raver22]

I can provide a general approach to making a matrix positive definite. First, let's define what a positive definite matrix is. A positive definite matrix is a square matrix where all of its eigenvalues are positive. This means that when you multiply the matrix by any non-zero vector, the resulting vector will have a positive length.

Now, to make a matrix positive definite, we can use either the singular value decomposition (SVD) or eigenvalue decomposition. Both methods involve finding the eigenvalues of the matrix, which are the values that satisfy the equation Av = λv, where A is the matrix, v is a non-zero vector, and λ is the eigenvalue.

If we use SVD, we can decompose the matrix into three matrices: U, Σ, and V. Here, U and V are orthogonal matrices and Σ is a diagonal matrix with the singular values of A on its diagonal. To make the matrix positive definite, we can simply replace any negative singular values on Σ with their absolute values. This will result in a new matrix that is positive definite.

Alternatively, we can use eigenvalue decomposition, which decomposes the matrix into two matrices: P and D. P is an orthogonal matrix and D is a diagonal matrix with the eigenvalues of A on its diagonal. To make the matrix positive definite, we can replace any negative eigenvalues on D with their absolute values. This will also result in a new matrix that is positive definite.

In summary, to make a matrix positive definite, we can use either SVD or eigenvalue decomposition and replace any negative singular values or eigenvalues with their absolute values. This will result in a new matrix that satisfies the definition of a positive definite matrix.