A totally nonnegative matrix is a real matrix every minor of which is nonnegative.
Thus every square submatrix has nonnegative determinant, including the 
submatrices consisting
of individual entries (Fallat and Johnson 2011). This is stronger than requiring
only nonnegative entries.
For a 
matrix, total nonnegativity is equivalent to nonnegative
entries and nonnegative determinant.
For example, the 
identity matrix
is totally nonnegative but is not a totally
positive matrix, whose minors must all be strictly
positive. In contrast, the matrix
with rows 
and 
has positive entries but determinant 
,
so it is not totally nonnegative.
For a square totally nonnegative matrix, the normalized immanants associated with the partitions 
decrease from the permanent
to the determinant as 
decreases from 
to 1 (Skandera 2026, p. 16).
