Numerical comparisons of matrices

Question

I have a matrix which should be equal to a null matrix. However due to the numerical precision, a brutal equality test with a matrix initialized with zeros does not work.

How should I perform the numerical equality test (with a given threshold for the precision) ?

Answer 1

A simpler way than to adjust the threshold for Equal is to use Chop which

replaces approximate real numbers in expr that are close to zero by the exact integer 0.

Adding the suggestions from the comments you have the following possibilities:

• Use Chop as it is. Here, you may only chop the Norm at the end by Chop[Norm[mat, 1]] == 0.
• Look at the second argument to Chop when you want to adjust the default tolerance. Ideally, it should correspond to the "sizes" of the matrices from which the putative null was constructed. For instance, if those matrices involve numbers in the millions, then any matrix whose coordinates are all less than about 0.0001 would likely need to be considered null. Typically, the second argument will be somewhere between the smallest and largest absolute values of the eigenvalues of those matrices. (These eigenvalues can reliably be found with SingularValueDecomposition; in many applications, they are already available from previous computations.)
• Look at the Internal`$EqualTolerance (which is probably not the best idea in your case).