A Kahan matrix is an upper triangular matrix used to test matrix rank estimation from a QR
decomposition (Kahan 1966, pp. 791-792). An unperturbed 
version has entries
 |
(1)
|
where 
, 
, and 
.
Each column has unit Euclidean norm. Thus column choices
in a pivoted QR decomposition can leave the columns
in their original order, with 
and 
as a factorization. The last diagonal
entry can substantially overestimate the smallest singular
value. Implementations may add a small diagonal
matrix to preserve the intended column order in floating-point
arithmetic (Higham 1995, pp. 30-31 and 49).
The plot shows 
for the unperturbed family with 
. Its growth illustrates
why the size of a diagonal entry alone need not reveal
the smallest singular value.
