Properties
- The above definition is equivalent to the statement
- A matrix is a Monge array if and only if for all and .
- Any subarray produced by selecting certain rows and columns from an original Monge array will itself be a Monge array.
- Any linear combination with non-negative coefficients of Monge arrays is itself a Monge array.
- One interesting property of Monge arrays is that if you mark with a circle the leftmost minimum of each row, you will discover that your circles march downward to the right; that is to say, if, then for all . Symmetrically, if you mark the uppermost minimum of each column, your circles will march rightwards and downwards. The row and column maxima march in the opposite direction: upwards to the right and downwards to the left.
- The notion of weak Monge arrays has been proposed; a weak Monge array is a square n-by-n matrix which satisfies the Monge property only for all .
Read more about this topic: Monge Array
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