I have the expression for the rank x i of a page i in an internet with n sites, each site contains n i links to other sites and is linked to by the pages L i ⊂ { 1 , … , n }. The expression is: (1) x i = ∑ j ∈ L i x j n j We can express this in matrix form as: x i = ∑ j ∈ L i x j n j ⟹ x i − ∑ j ∈ L i x j n j = 0And therefore we can write the system of linear equations as: A x → = 0 → , with: A i j = { 1 i = j − 1 n j j ∈ L i 0 j ∉ L i ∪ { i } I have two questions:1. Is there a better way to write the components of the matrix which doesn't involve a conditional expression?2. Can we show that there always ∃ x → ∣ x → ≠ 0 → ?

Question

I have the expression for the rank       x      i   of a page   i in an internet with   n sites, each site contains       n      i   links to other sites and is linked to by the pages       L      i    ⊂  {  1  ,  …  ,  n  }. The expression is:                    (1)                              x                  i                =                  ∑                      j            ∈                          L                          i                                                            x                      j                                n                      j                              We can express this in matrix form as:      x      i    =      ∑          j      ∈              L              i                        x          j              n          j          ⟹        x      i    −      ∑          j      ∈              L              i                        x          j              n          j        =  0And therefore we can write the system of linear equations as:  A      x    →    =      0    →  , with:      A          i      j        =      {                            1                          i          =          j                                      −                      1                          n                          j                                                j          ∈                      L                      i                                                0                          j          ∉                      L                      i                    ∪          {          i          }                        I have two questions:1. Is there a better way to write the components of the matrix which doesn&#039;t involve a conditional expression?2. Can we show that there always   ∃      x    →    ∣      x    →    ≠      0    →  ?

kpgt1z · Accepted Answer

(1) I am not aware of a better way to rewrite the matrix entries.(2) The matrix   A is a stochastic matrix: Let   e  =  (  1  ,  …  ,  1      )    T   be the columns vector filled with ones. Then      A    T    e  =  0.Thus   A has not full rank, and there must be non-zero vectors such that  A  x  =  0.

tr2os8x · Accepted Answer

I have been looking for an answer to this question for a very long time, thank you very much!

I have the expression for the rank x i </msub> of a page i in an interne

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