Why does chol([4, -4;-4, 4]) fail to produce an answer?

>> chol([4, -4;-4, 4])

Error using chol

Matrix must be positive definite.

Yet clearly

R = [2, -2];

R' * R

Gives ans =

     4    -4
    -4     4

Answer 1

[Stephen23]

Actually that matrix is not positive definite. From the chol documentation:

" [R,p] = chol(A) for positive definite A, produces an upper triangular matrix R from the diagonal and upper triangle of matrix A, satisfying the equation R'*R=A and p is zero. If A is not positive definite, then p is a positive integer and MATLAB® does not generate an error."

Lets try it:

>> [~,p] = chol([4, -4;-4, 4])
p =
     2

This is the same method that TMW gives here:

http://www.mathworks.com/matlabcentral/answers/101132-how-do-i-determine-if-a-matrix-is-positive-definite-using-matlab

[Paul Mellor]

I see. Then is there a way to obtain a factorization R' * R = A for this matrix, even though it is not defined to be positive definite (since one of its eigenvalues is zero)? This factorization clearly exists as R = [2, -2];