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I've been trying to show that the set of matrices that preserve L1 norm (sum of absolute values of each coordinate) are the complex permutation matrices. Complex permutation matrix is defined as permutation of the columns of complex diagonal matrix with magnitude of each diagonal element equal 1. (This problem is from Matrix Analysis by Horn and Johnson).

I've spend quite a lot of time trying to solve this but without much success. It is easy to show that complex permutation matrix keeps the L1 norm of a vector the same, but I can't show the opposite, that every such matrix must be a complex permutation matrix. The only thing I came up with is to take a vector [tex]x=(1-\epsilon,\epsilon)[/tex], and just say that L1 norm of [tex]Bx[/tex] can't be linear for all values of [tex]\epsilon \ge 1[/tex]. But that is not really a proof.

Also it is easy to show from scalar product that such set of transformations for L2 norm are the unitary matrices. If I could somehow show that if a matrix preserves L1 norm then it must necessarily preserve L2 norm, then norm preserving matrices for L1 must be a subset of norm preserving matrices for L2. Then it is fairly easy to show that such a unitary matrix must be complex permutation matrix. But I can't seem to show that L1 preserving matrix must be L2 preserving.

Please, if someone could give me some hints or suggestions, it would be very appreciated!

Thanks.

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# Isometry group for L1 norm

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