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Chessboard distance between matrices

Posted 3 months ago
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Hello, I'm currently studying the Chebyshev distance (chessboard distance). The distance between two points x, y, with coordinates xi, yi, is maxi (|xi - y_i|). I'm interested in evaluating the distance between two matrices. In Mathematica, ChessboardDistance(MatrixA, MatrixA) is zero, as expected, because element-wise differences are all zero. ChessboardDistance(MatrixA, MatrixB) seems to give the max of the (sum in each row) of abs values of element-wise differences. Is that correct? I'm still looking for a reference with the formula for Chebyshev distance between two matrices. Example: MatrixA = {{1, 0, -1}, {2, 1, 0}, {0, 0, 1}}; MatrixB = {{0, 0, 4}, {-1, 3, 2}, {0, 0, 1}}; MatrixB- MatrixA = {{-1, 0, 5}, {-3, 2, 2}, {0, 0, 0}}. ChessboardDistance[MatrixC, MatrixB] = 7, that is, looking at the second row, abs (-3) + abs (2) + abs (2) = 3 + 2 + 2 = 7. Thank you for the information you can give me :-)

2 Replies

That is correct.

For matrices that distance is treated as the operator norm. By definition, for matrix m this is the maximum value, among all vectors v of Chebyshev norm 1, of ||m.v||. Since we can make every element of v have absolute value of 1, this is seen to be that maximum row sum of absolute values.

This might in general be a better question for a math forum, by the way. It's not specific to Mathematica.

Posted 3 months ago

Thank you so much! :)

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