>  # mxRex09.r   
>  #
>  # my solutions to 'Some Matrix Exercises in R'
>  #
>  # (1) Create X
>   n <- 7            # arbitrary size
>  #
>   X <- matrix(c(rep(1,n),(1:n),(1:n)*(1:n)),n,3)
>   X
     [,1] [,2] [,3]
[1,]    1    1    1
[2,]    1    2    4
[3,]    1    3    9
[4,]    1    4   16
[5,]    1    5   25
[6,]    1    6   36
[7,]    1    7   49
>  #  changing to arbitrary degree is harder
>   diff(diff(X))
     [,1] [,2] [,3]
[1,]    0    0    2
[2,]    0    0    2
[3,]    0    0    2
[4,]    0    0    2
[5,]    0    0    2
>  #
>  # (2) Create A
>   A <- matrix(c(1,-2,1,rep(0,n-2)),n-2,n,byrow=TRUE)
>   A
     [,1] [,2] [,3] [,4] [,5] [,6] [,7]
[1,]    1   -2    1    0    0    0    0
[2,]    0    1   -2    1    0    0    0
[3,]    0    0    1   -2    1    0    0
[4,]    0    0    0    1   -2    1    0
[5,]    0    0    0    0    1   -2    1
>  #     Multiply AX
>   A %*% X
     [,1] [,2] [,3]
[1,]    0    0    2
[2,]    0    0    2
[3,]    0    0    2
[4,]    0    0    2
[5,]    0    0    2
>  #
>  #
>  # (3) Toeplitz matrix
>  # first way
>   Z <- matrix(0,n,n)          # create zero matrix 
>   E <- abs( row(Z) - col(Z) )
>   V <- log(1+E)
>   V
          [,1]      [,2]      [,3]      [,4]      [,5]      [,6]      [,7]
[1,] 0.0000000 0.6931472 1.0986123 1.3862944 1.6094379 1.7917595 1.9459101
[2,] 0.6931472 0.0000000 0.6931472 1.0986123 1.3862944 1.6094379 1.7917595
[3,] 1.0986123 0.6931472 0.0000000 0.6931472 1.0986123 1.3862944 1.6094379
[4,] 1.3862944 1.0986123 0.6931472 0.0000000 0.6931472 1.0986123 1.3862944
[5,] 1.6094379 1.3862944 1.0986123 0.6931472 0.0000000 0.6931472 1.0986123
[6,] 1.7917595 1.6094379 1.3862944 1.0986123 0.6931472 0.0000000 0.6931472
[7,] 1.9459101 1.7917595 1.6094379 1.3862944 1.0986123 0.6931472 0.0000000
>  # second way
>   E <- abs( matrix(1:n,n,n) - matrix(1:n,n,n,byrow=T) )
>   V <- log(1+E)
>   V
          [,1]      [,2]      [,3]      [,4]      [,5]      [,6]      [,7]
[1,] 0.0000000 0.6931472 1.0986123 1.3862944 1.6094379 1.7917595 1.9459101
[2,] 0.6931472 0.0000000 0.6931472 1.0986123 1.3862944 1.6094379 1.7917595
[3,] 1.0986123 0.6931472 0.0000000 0.6931472 1.0986123 1.3862944 1.6094379
[4,] 1.3862944 1.0986123 0.6931472 0.0000000 0.6931472 1.0986123 1.3862944
[5,] 1.6094379 1.3862944 1.0986123 0.6931472 0.0000000 0.6931472 1.0986123
[6,] 1.7917595 1.6094379 1.3862944 1.0986123 0.6931472 0.0000000 0.6931472
[7,] 1.9459101 1.7917595 1.6094379 1.3862944 1.0986123 0.6931472 0.0000000
>  #
>  # (4) V(i,j) = max(i,j)
>  #  similar to previous approach
>   R <- matrix(1:n,n,n)
>   C <- matrix(1:n,n,n,byrow=T) 
>   V <- ( (R+C) + abs(R-C) )/2
>   V
     [,1] [,2] [,3] [,4] [,5] [,6] [,7]
[1,]    1    2    3    4    5    6    7
[2,]    2    2    3    4    5    6    7
[3,]    3    3    3    4    5    6    7
[4,]    4    4    4    4    5    6    7
[5,]    5    5    5    5    5    6    7
[6,]    6    6    6    6    6    6    7
[7,]    7    7    7    7    7    7    7
>  # try another way
>   V <- pmax(R,C)
>   V
     [,1] [,2] [,3] [,4] [,5] [,6] [,7]
[1,]    1    2    3    4    5    6    7
[2,]    2    2    3    4    5    6    7
[3,]    3    3    3    4    5    6    7
[4,]    4    4    4    4    5    6    7
[5,]    5    5    5    5    5    6    7
[6,]    6    6    6    6    6    6    7
[7,]    7    7    7    7    7    7    7
>  # (5) J(i,j) = 1 only where i+j=n+1
>   J <- diag(rep(1,n))[rev(1:n),]
>   J
     [,1] [,2] [,3] [,4] [,5] [,6] [,7]
[1,]    0    0    0    0    0    0    1
[2,]    0    0    0    0    0    1    0
[3,]    0    0    0    0    1    0    0
[4,]    0    0    0    1    0    0    0
[5,]    0    0    1    0    0    0    0
[6,]    0    1    0    0    0    0    0
[7,]    1    0    0    0    0    0    0
>   J <- diag(1,n)[rev(1:n),]
>   J
     [,1] [,2] [,3] [,4] [,5] [,6] [,7]
[1,]    0    0    0    0    0    0    1
[2,]    0    0    0    0    0    1    0
[3,]    0    0    0    0    1    0    0
[4,]    0    0    0    1    0    0    0
[5,]    0    0    1    0    0    0    0
[6,]    0    1    0    0    0    0    0
[7,]    1    0    0    0    0    0    0
>  #
>  # (6) K = 2x? matrix of all combinations
>   K <- matrix(c(rep((1:3),4),rep((1:4),each=3)),nrow=2,byrow=TRUE)
>   K
     [,1] [,2] [,3] [,4] [,5] [,6] [,7] [,8] [,9] [,10] [,11] [,12]
[1,]    1    2    3    1    2    3    1    2    3     1     2     3
[2,]    1    1    1    2    2    2    3    3    3     4     4     4
>  #
>  # done -- clean up 
>    rm(list=ls())
>    q()