[Zhan Xiang matrix theory exercise reference] exercise 1.14

14. if ing $ F: M_n \ To M_n $ sorts the elements of each matrix in $ M_n $ in a fixed mode, $ F $ is called a replacement operator. what kind of replacement operator does not change the feature value of the Matrix? Keep the rank unchanged?

 

 

Answer: The replacement operator $ F $ keeps the feature value of the matrix unchanged. If only the replacement matrix $ p $ exists, $ \ Bex F (a) = pap ^ t, \ quad \ forall \ A \ In M_n; \ EEx $ or $ \ Bex F (a) = PA ^ TP ^ t, \ quad \ forall \ A \ In M_n. \ EEx $ replacement operator $ F $ keep the Singular Values of the matrix unchanged if and only if there is a replacement matrix $ p $, $ q $, making $ \ Bex F () = PAQ, \ quad \ forall \ A \ In M_n; \ EEx $ or $ \ Bex F (a) = PA ^ TQ, \ quad \ forall \ A \ In M_n. \ EEx $

[Zhan Xiang matrix theory exercise reference] exercise 1.14