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Which conditions must the matrix entries satisfy, and what would be an interpretation of the row and column sums of the matrix?

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Hint: Such an isometry would have to map the $l^1$-unit-sphere onto itself. How does this "sphere" look like?

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    Ok, am I correct if I argue that a linear isometry is characterized by that it takes extreme points to extreme points, and, since a) the extreme points in this case are characterized by $\|x\|_\infty=1$, and b) $\|Ax\|_\infty$ is just the absolute column maximum for one of the columns of A for any extreme point, it follows that A must consists of zeros and ones, with exactly one 1 in each column and have full rank, i.e. A must be a permutation matrix?2011-01-20