> # MATRICES
> 
> x<-c(1,3,2,10,5)    #create a vector x with 5 components 
> y<-1:5              #create a vector of consecutive integers 
> x
[1]  1  3  2 10  5
> y
[1] 1 2 3 4 5
> m1<-cbind(x,y)    #Column bind
> m1
      x y
[1,]  1 1
[2,]  3 2
[3,]  2 3
[4,] 10 4
[5,]  5 5
> t(m1)                # transpose of m1 
  [,1] [,2] [,3] [,4] [,5]
x    1    3    2   10    5
y    1    2    3    4    5
> dim(m1)  
[1] 5 2
> m1<-rbind(x,y)       # rbind() is for row bind and equivalent to t(cbind()).
> m1
  [,1] [,2] [,3] [,4] [,5]
x    1    3    2   10    5
y    1    2    3    4    5
> 
> # LIST ELEMENTS AND SPECIFY MATRIX DIRECTLY (WITHOUT BINDING)
> 
> m2<-matrix(c(1,3,2,5,-1,2,2,3,9),nrow=3)
> m2
     [,1] [,2] [,3]
[1,]    1    5    2
[2,]    3   -1    3
[3,]    2    2    9
> 
> m2<-matrix(c(1,3,2,5,-1,2,2,3,9),ncol=3,byrow=T)    # to fill row-wise
> m2
     [,1] [,2] [,3]
[1,]    1    3    2
[2,]    5   -1    2
[3,]    2    3    9
> 
> m2[2,3]            #element of m2 at the second row, third column 
[1] 2
> m2[2,]             #second row 
[1]  5 -1  2
> m2[,3]             #third column 
[1] 2 2 9
> m2[-1,]            #submatrix of m2 without the first row 
     [,1] [,2] [,3]
[1,]    5   -1    2
[2,]    2    3    9
> m2[,-1]            #without the first column
     [,1] [,2]
[1,]    3    2
[2,]   -1    2
[3,]    3    9
> m2[-1,-1]          #submatrix of m2 with the first row and column removed 
     [,1] [,2]
[1,]   -1    2
[2,]    3    9
> 
> # COMPONENT-WISE COMPUTATIONS
> 
> m1<-matrix(1:4, ncol=2); m2<-matrix(c(10,20,30,40),ncol=2) 
> m1
     [,1] [,2]
[1,]    1    3
[2,]    2    4
> m2
     [,1] [,2]
[1,]   10   30
[2,]   20   40
> 2*m1                # scalar multiplication 
     [,1] [,2]
[1,]    2    6
[2,]    4    8
> m1+m2               # matrix addition 
     [,1] [,2]
[1,]   11   33
[2,]   22   44
> m1*m2               # component-wise multiplication 
     [,1] [,2]
[1,]   10   90
[2,]   40  160
> 
> # MATRIX OPERATIONS
> 
> m1 %*% m2           # the usual matrix multiplication
     [,1] [,2]
[1,]   70  150
[2,]  100  220
> solve(m1)            #inverse matrix of m1 
     [,1] [,2]
[1,]   -2  1.5
[2,]    1 -0.5
> solve(m1)%*%m1       #check
     [,1] [,2]
[1,]    1    0
[2,]    0    1
> diag(3)              #diag() is used to construct a k by k identity matrix 
     [,1] [,2] [,3]
[1,]    1    0    0
[2,]    0    1    0
[3,]    0    0    1
> diag(c(2,3,3))       #as well as other diagonal matrices 
     [,1] [,2] [,3]
[1,]    2    0    0
[2,]    0    3    0
[3,]    0    0    3
>
> # BE CAREFUL.  R tries to read minds.
> diag(m2)
[1] 10 40
> 
> eigen(m2)           # eigenvalues and eigenvectors of m2
$values
[1] 53.722813 -3.722813

$vectors
           [,1]       [,2]
[1,] -0.5657675 -0.9093767
[2,] -0.8245648  0.4159736

> ev <- eigen(m2)     # ev is a variable with two components
> ev
$values
[1] 53.722813 -3.722813

$vectors
           [,1]       [,2]
[1,] -0.5657675 -0.9093767
[2,] -0.8245648  0.4159736

> ev$values           # to extract components, use $
[1] 53.722813 -3.722813
> ev$vectors
           [,1]       [,2]
[1,] -0.5657675 -0.9093767
[2,] -0.8245648  0.4159736
> attach(ev)             # to keep from typing ev
> values
[1] 53.722813 -3.722813
> vectors
           [,1]       [,2]
[1,] -0.5657675 -0.9093767
[2,] -0.8245648  0.4159736