# Perfect matrix

(diff) ← Previous revision | Latest revision (diff) | Newer revision → (diff)

In mathematics, a perfect matrix is an m-by-n binary matrix that has no possible k-by-k submatrix K that satisfies the following conditions:[1]

• k > 3
• the row and column sums of K are each equal to b, where b ≥ 2
• there exists no row of the (m − k)-by-k submatrix formed by the rows not included in K with a row sum greater than b.

The following is an example of a K submatrix where k = 5 and b = 2:

${\displaystyle {\begin{bmatrix}1&1&0&0&0\\0&1&1&0&0\\0&0&1&1&0\\0&0&0&1&1\\1&0&0&0&1\end{bmatrix}}.}$

## References

1. ^ D. M. Ryan, B. A. Foster, An Integer Programming Approach to Scheduling, p.274, University of Auckland, 1981.