Alternating n-linear function and matrices
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Exercise:
Let D be an n-linear function. Ase that D has the property that whenever A has two equal adjacent rows then DA. Then abcliste abc D is alternating. abc forall matrix B if B' is obtained from B by erchanging two of its rows then DB'-DB. abcliste
Solution:
Proof. We with the proof of b. Ase first that B' is obtained from B by erchanging two adjacent rows: B pmatrix hdots & beta_ & hdots hdots & vdots & hdots hdots & beta_k & hdots hdots & beta_k+ & hdots hdots & vdots & hdots hdots & beta_n & hdots pmatrix B' pmatrix hdots & beta_ & hdots hdots & vdots & hdots hdots & beta_k+ & hdots hdots & beta_k & hdots hdots & vdots & hdots hdots & beta_n & hdots pmatrix. Consider Dbeta_...beta_k+beta_k+beta_k+beta_k+...beta_n. On the other hand by n-linearity we have: Dbeta_...beta_kbeta_k+beta_k+...beta_n+Dbeta_...beta_k+beta_k+beta_k+...beta_n Dbeta_...beta_kbeta_k...beta_n+Dbeta_...beta_kbeta_k+...beta_n+Dbeta_...beta_k+beta_k...beta_n+Dbeta_...beta_k+beta_k+...beta_n &Longrightarrow Dbeta_...beta_kbeta_k+...beta_n+Dbeta_...beta_k+beta_k...beta_n Now suppose that B' is obtained from B by erchanging rows kl where k l not necessarily lk+. We can obtain B' from B by doing a sequence of erchanges of pairs of adjacent rows. For example we can with rows k and k+ and continue in this manner till we get beta_...beta_k-beta_k+...beta_lbeta_k beta_l+...beta_n. This requires rl-k erchanges of adjacent rows. Next we move beta_l to position k by doing r- erchanging of adjacent rows and we arrive to B'. The total number of erchanes of adjacent rows is r+r-r- Longrightarrow DB'-^r-DB-DB. This completes the proof of b. We prove a. Let A be a matrix in which row i equals to row j where i j. Denote the rows of A by B pmatrix hdots & alpha_ & hdots hdots & vdots & hdots hdots & alpha_i & hdots hdots & alpha_i+ & hdots hdots & vdots & hdots hdots & alpha_j & hdots hdots & vdots & hdots hdots & alpha_n & hdots pmatrix. If ji+ then by asption DA. If j i+ we erchange alpha_i+ with alpha_j and we obtain a matrix A' with two equal adjacent rows. By asption DA'. But by b DA'-DA Longrightarrow DA.
Let D be an n-linear function. Ase that D has the property that whenever A has two equal adjacent rows then DA. Then abcliste abc D is alternating. abc forall matrix B if B' is obtained from B by erchanging two of its rows then DB'-DB. abcliste
Solution:
Proof. We with the proof of b. Ase first that B' is obtained from B by erchanging two adjacent rows: B pmatrix hdots & beta_ & hdots hdots & vdots & hdots hdots & beta_k & hdots hdots & beta_k+ & hdots hdots & vdots & hdots hdots & beta_n & hdots pmatrix B' pmatrix hdots & beta_ & hdots hdots & vdots & hdots hdots & beta_k+ & hdots hdots & beta_k & hdots hdots & vdots & hdots hdots & beta_n & hdots pmatrix. Consider Dbeta_...beta_k+beta_k+beta_k+beta_k+...beta_n. On the other hand by n-linearity we have: Dbeta_...beta_kbeta_k+beta_k+...beta_n+Dbeta_...beta_k+beta_k+beta_k+...beta_n Dbeta_...beta_kbeta_k...beta_n+Dbeta_...beta_kbeta_k+...beta_n+Dbeta_...beta_k+beta_k...beta_n+Dbeta_...beta_k+beta_k+...beta_n &Longrightarrow Dbeta_...beta_kbeta_k+...beta_n+Dbeta_...beta_k+beta_k...beta_n Now suppose that B' is obtained from B by erchanging rows kl where k l not necessarily lk+. We can obtain B' from B by doing a sequence of erchanges of pairs of adjacent rows. For example we can with rows k and k+ and continue in this manner till we get beta_...beta_k-beta_k+...beta_lbeta_k beta_l+...beta_n. This requires rl-k erchanges of adjacent rows. Next we move beta_l to position k by doing r- erchanging of adjacent rows and we arrive to B'. The total number of erchanes of adjacent rows is r+r-r- Longrightarrow DB'-^r-DB-DB. This completes the proof of b. We prove a. Let A be a matrix in which row i equals to row j where i j. Denote the rows of A by B pmatrix hdots & alpha_ & hdots hdots & vdots & hdots hdots & alpha_i & hdots hdots & alpha_i+ & hdots hdots & vdots & hdots hdots & alpha_j & hdots hdots & vdots & hdots hdots & alpha_n & hdots pmatrix. If ji+ then by asption DA. If j i+ we erchange alpha_i+ with alpha_j and we obtain a matrix A' with two equal adjacent rows. By asption DA'. But by b DA'-DA Longrightarrow DA.