15.1 Distance on (U, U)
When dealing with matrices, it is natural to say that that the matrix
is close to the 2 × 2 identity matrix I. On the other hand, if we deal with 104 × 104 matrices, we may suspect that the matrix B given by bii = 1 and bij = 10–3 for i ≠ j behaves very differently from the identity matrix. It is natural to seek a notion of distance between n × n matrices which would measure closeness in some useful way.
What do we wish to mean by saying that we want to ‘measure closeness in some useful way’? Surely, we ...