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.
