The gap distance to the set of singular matrix pencils
A new distance measure for matrix pencils based on the gap distance between subspaces is introduced. Its properties are discussed and studied. The authors provide upper and lower bounds for the gap distance to singularity and a comparison of the latter with the distance to singularity defined in terms of the Frobenius norm is given. It is found that the main difference is that the gap distance is invariant under multiplications of the pencil from the left with an arbitrary invertible matrix. Some examples show that the gap distance to singularity may be used as tool to detect whether a small distance to singularity is a result of bad scaling of a matrix pencil in the following way: if the gap distance to singularity is also small, then probably the pencil is indeed close to a singular pencil (at least with respect to these two distance measures). If, however, the gap distance to singularity is not equally small, then this suggests that the small distance to singularity is only due to bad scaling of the matrix pencil.
scientific article; zbMATH DE number 7007590
scientific article