### Matrices, Lists, Factors

#### Matrices:

R can also be used for matrix calculations. Matrices have rows and columns containing a single data type. In a matrix, the order of rows and columns is important.

m <- matrix(nrow = 2, ncol = 3)
dim(m)

attributes(m)
m <- matrix(1:20, nrow = 4, ncol = 5)
m

Output:

> dim(m)
[1] 2 3
>
> attributes(m)
\$dim
[1] 2 3

> m <- matrix(1:20, nrow = 4, ncol = 5)
> m
[,1] [,2] [,3] [,4] [,5]
[1,] 1 5 9 13 17
[2,] 2 6 10 14 18
[3,] 3 7 11 15 19
[4,] 4 8 12 16 20

Matrices can also be created directly from vectors by adding a dimension attribute.

m <- 1:20
m
dim(m) <- c(4, 5)
m

Output:

> m
[1] 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20
> dim(m) <- c(4, 5)
> m
[,1] [,2] [,3] [,4] [,5]
[1,] 1 5 9 13 17
[2,] 2 6 10 14 18
[3,] 3 7 11 15 19
[4,] 4 8 12 16 20

Matrices can be created by column-binding or row-binding with the cbind() and rbind() functions.

x<-1:3
y<-10:12
z<-30:32
cbind(x,y,z)
rbind(x,y,z)

Output:

> cbind(x,y,z)
x y z
[1,] 1 10 30
[2,] 2 11 31
[3,] 3 12 32
> rbind(x,y,z)
[,1] [,2] [,3]
x 1 2 3
y 10 11 12
z 30 31 32

By default the matrix function reorders a vector into columns, but we can also tell R to use rows instead.

x <-1:9
matrix(x, nrow = 3, ncol = 3)
matrix(x, nrow = 3, ncol = 3, byrow = TRUE)

Output:

> cbind(x,y,z)
x y z
[1,] 1 10 30
[2,] 2 11 31
[3,] 3 12 32
> rbind(x,y,z)
[,1] [,2] [,3]
x 1 2 3
y 10 11 12
z 30 31 32

We can also create a matrix of a specified dimension where every element is the same.

z<- matrix(5, 3, 4)
z

Output:

> z
[,1] [,2] [,3] [,4]
[1,] 5 5 5 5
[2,] 5 5 5 5
[3,] 5 5 5 5

We can create a matrix with specified elements on the diagonal. (And 0 on the off-diagonals.)

diag(3)
diag(1:4)

Output:

> diag(3)
[,1] [,2] [,3]
[1,] 1 0 0
[2,] 0 1 0
[3,] 0 0 1
> diag(1:4)
[,1] [,2] [,3] [,4]
[1,] 1 0 0 0
[2,] 0 2 0 0
[3,] 0 0 3 0
[4,] 0 0 0 4

Like vectors, matrices can be subsetted using square brackets, []. However, since matrices are two dimensional, we need to specify both a row and a column when subsetting.

Here we accessed the element in the first row and the second column.

z<- matrix(1:12, 3, 4)
z

Output:

> z
[,1] [,2] [,3] [,4]
[1,] 1 4 7 10
[2,] 2 5 8 11
[3,] 3 6 9 12

We could also subset an entire row.

z[1, ]

Output:

[1]  1  4  7 10

We could also subset an entire column.

z[ ,2]

Output:

[1] 4 5 6

We can also use vectors to subset more than one row or column at a time. Here we subset to the first and third column of the second row.

z[2, c(1, 3)]

Output:

[1] 2 8

#### Matrix Operations:

x = 1:9
y = 9:1
x = matrix(x, 3, 3)
y = matrix(y, 3, 3)
x
x + y
x-y
x*y
x/y

Output:

> x
[,1] [,2] [,3]
[1,] 1 4 7
[2,] 2 5 8
[3,] 3 6 9
> x + y
[,1] [,2] [,3]
[1,] 10 10 10
[2,] 10 10 10
[3,] 10 10 10
> x-y
[,1] [,2] [,3]
[1,] -8 -2 4
[2,] -6 0 6
[3,] -4 2 8
> x*y
[,1] [,2] [,3]
[1,] 9 24 21
[2,] 16 25 16
[3,] 21 24 9
> x/y
[,1] [,2] [,3]
[1,] 0.1111111 0.6666667 2.333333
[2,] 0.2500000 1.0000000 4.000000
[3,] 0.4285714 1.5000000 9.000000

Note that X * Y is not matrix multiplication. It is element by element multiplication. (Same for X / Y). Instead, matrix multiplication uses %*%. Other matrix functions include t() which gives the transpose of a matrix and solve() which returns the inverse of a square matrix if it is invertible.

x%*%y
t(x)

z<-matrix(c(9, 2, -3, 2, 4, -2, -3, -2, 16), 3, byrow = TRUE)
solve(z)

z<-matrix(c(9, 2, -3, 2, 4, -2, -3, -2, 16), 3, byrow = TRUE)
dim(z)
rowSums(z)
colSums(z)
rowMeans(z)
colMeans(z)
diag(z)

Output:

> x%*%y
[,1] [,2] [,3]
[1,] 90 54 18
[2,] 114 69 24
[3,] 138 84 30
> t(x)
[,1] [,2] [,3]
[1,] 1 2 3
[2,] 4 5 6
[3,] 7 8 9
>
> z<-matrix(c(9, 2, -3, 2, 4, -2, -3, -2, 16), 3, byrow = TRUE)
> solve(z)
[,1] [,2] [,3]
[1,] 0.12931034 -0.05603448 0.01724138
[2,] -0.05603448 0.29094828 0.02586207
[3,] 0.01724138 0.02586207 0.06896552
> z<-matrix(c(9, 2, -3, 2, 4, -2, -3, -2, 16), 3, byrow = TRUE)
> dim(z)
[1] 3 3
> rowSums(z)
[1] 8 4 11
> colSums(z)
[1] 8 4 11
> rowMeans(z)
[1] 2.666667 1.333333 3.666667
> colMeans(z)
[1] 2.666667 1.333333 3.666667
> diag(z)
[1] 9 4 16

#### Lists:

Lists are a special type of vector that can contain elements of different classes.

`x <- list("stat",5.1, TRUE, 1 + 4i)xclass(x)`

Output:

> x
[[1]]
[1] “stat”

[[2]]
[1] 5.1

[[3]]
[1] TRUE

[[4]]
[1] 1+4i

> class(x)
[1] “list”

You can  create an empty list of a prespecified length with the vector() function.

`x <- vector("list", length = 10)`

x

Output:

> x
[[1]]
NULL

[[2]]
NULL

[[3]]
NULL

[[4]]
NULL

[[5]]
NULL

[[6]]
NULL

[[7]]
NULL

[[8]]
NULL

[[9]]
NULL

[[10]]
NULL

We can create a little bit complex List like below.

`l <-list(a <- c(1, 2, 3, 4),b <- FALSE,c <- "Hello Statistics!",d = function(arg = 42) {print("Hello World!")},e = diag(10))`

l

Output:

> l
[[1]]
[1] 1 2 3 4

[[2]]
[1] FALSE

[[3]]
[1] “Hello Statistics!”

\$d
function (arg = 42)
{
print(“Hello World!”)
}

\$e
[,1] [,2] [,3] [,4] [,5] [,6] [,7] [,8] [,9] [,10]
[1,] 1 0 0 0 0 0 0 0 0 0
[2,] 0 1 0 0 0 0 0 0 0 0
[3,] 0 0 1 0 0 0 0 0 0 0
[4,] 0 0 0 1 0 0 0 0 0 0
[5,] 0 0 0 0 1 0 0 0 0 0
[6,] 0 0 0 0 0 1 0 0 0 0
[7,] 0 0 0 0 0 0 1 0 0 0
[8,] 0 0 0 0 0 0 0 1 0 0
[9,] 0 0 0 0 0 0 0 0 1 0
[10,] 0 0 0 0 0 0 0 0 0 1

Lists can be subset using two syntaxes, the \$ operator, and square brackets []. The \$ operator returns a named element of a list. The [] syntax returns a list, while the [[]] returns an element of a list.

`# subsettingl\$el["e"]l[1:2]`

Output:

> l\$e
[,1] [,2] [,3] [,4] [,5] [,6] [,7] [,8] [,9] [,10]
[1,] 1 0 0 0 0 0 0 0 0 0
[2,] 0 1 0 0 0 0 0 0 0 0
[3,] 0 0 1 0 0 0 0 0 0 0
[4,] 0 0 0 1 0 0 0 0 0 0
[5,] 0 0 0 0 1 0 0 0 0 0
[6,] 0 0 0 0 0 1 0 0 0 0
[7,] 0 0 0 0 0 0 1 0 0 0
[8,] 0 0 0 0 0 0 0 1 0 0
[9,] 0 0 0 0 0 0 0 0 1 0
[10,] 0 0 0 0 0 0 0 0 0 1
> l[“e”]
\$e
[,1] [,2] [,3] [,4] [,5] [,6] [,7] [,8] [,9] [,10]
[1,] 1 0 0 0 0 0 0 0 0 0
[2,] 0 1 0 0 0 0 0 0 0 0
[3,] 0 0 1 0 0 0 0 0 0 0
[4,] 0 0 0 1 0 0 0 0 0 0
[5,] 0 0 0 0 1 0 0 0 0 0
[6,] 0 0 0 0 0 1 0 0 0 0
[7,] 0 0 0 0 0 0 1 0 0 0
[8,] 0 0 0 0 0 0 0 1 0 0
[9,] 0 0 0 0 0 0 0 0 1 0
[10,] 0 0 0 0 0 0 0 0 0 1

> l[1:2]
[[1]]
[1] 1 2 3 4

[[2]]
[1] FALSE

#### Factor

Factors are used to represent categorical data and can be unordered or ordered. An example might be “Male” and “Female” if we consider gender. Factor objects can be created with the factor() function.

`x <- factor(c("male", "female", "male", "male", "female"))xtable(x)`

Output:

> x
[1] male female male male female
Levels: female male
> table(x)
x
female male
2 3

By default Levels are put in alphabetical order. If you print the above code you will get levels as female and male. But if you want to get your levels in particular order then set levels parameter like this.

`x <- factor(c("male", "female", "male", "male", "female"), levels=c("male", "female"))xtable(x)`

Output:

> x
[1] male female male male female
Levels: male female
> table(x)
x
male female
3 2