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
Famous quotes containing the word properties:
“A drop of water has the properties of the sea, but cannot exhibit a storm. There is beauty of a concert, as well as of a flute; strength of a host, as well as of a hero.”
—Ralph Waldo Emerson (1803–1882)
“The reason why men enter into society, is the preservation of their property; and the end why they choose and authorize a legislative, is, that there may be laws made, and rules set, as guards and fences to the properties of all the members of the society: to limit the power, and moderate the dominion, of every part and member of the society.”
—John Locke (1632–1704)