# Matrix in R

In this tutorial, you will learn about matrix data type in R. We will discuss about how to create a matrix in R, how to access subset of a matrix and how to perform various operations like addition, multiplication, transpose, inverse, determinant and eigen values on matrix.

## Matrix in R

A matrix is a two-dimensional data structure in R. It is a set of elements appearing in rows and columns. Matrix in R is atomic (homogeneous) data structure. That is all the elements of the matrix must be of same mode (logical, numeric, complex or character).

A matrix can be created with the function matrix:

matrix(data=NA, nrow=r, ncol=c, byrow=FALSE, dimnames=NULL)

• By default matrices are created with their values running down successive column.
• It is also possible to specify that the matrix be filled by rows using byrow=TRUE.
• The option dimnames allows to give names to the rows and columns.

### Attributes related to matrix (or data)

Some useful functions related to matrices in R are as follows:

Function Description
class(x) Gives the class of object x as matrix
typeof(x) Gives the type of data stored in matrix
dim(x) Sets or changes the dimension of x
str(x) Gives the structure of a matrix
nrow(x) Gives the number of rows of a matrix
ncol(x) Gives the number of columns of a matrix
rownames(x) Gives row names of a matrix
colnames(x) Gives column names of a matrix

## How to Create matrix in R?

### Creating an empty matrix

An empty matrix can be created just by specifying the number of rows and number of columns of a matrix.

AE <- matrix(nrow=3,ncol=4)

Above command create an empty matrix (with elements NA) of dimension 3 by 4 and store it in AE.

AE
     [,1] [,2] [,3] [,4]
[1,]   NA   NA   NA   NA
[2,]   NA   NA   NA   NA
[3,]   NA   NA   NA   NA

The dim(x) function either display or set the dimension of object x.

# display the dimension of given matrix
dim(AE)  
[1] 3 4

The nrow(x) and ncol(x) display the number of rows and number of columns present in object x.

# gives the number of rows in a given matrix
nrow(AE)  
[1] 3
# gives the number of columns in a given matrix
ncol(AE)  
[1] 4

The class(x) function display the class of object x. The object AE is defined as a matrix, so the class(AE) function display the class as matrix.

class(AE)
[1] "matrix" "array" 

In R, NA are the logical constants of length 1. So the typeof() an empty matrix shows logical.

typeof(AE) 
[1] "logical"

The str(x) display the structure of object x.

str(AE)
 logi [1:3, 1:4] NA NA NA NA NA NA ...

The object AE is a matrix, so the str(AE) function display the structure of AE with type as logical, number of rows 1:3, number of columns 1:4 and few elements of the object AE.

### Creating Zero matrix

Zero matrix can be created by specifying the element 0 and the number of rows and columns of a matrix.

Below R command create a zero matrix with 3 rows and 4 columns.

A0 <- matrix(0, nrow = 3, ncol = 4)
A0
     [,1] [,2] [,3] [,4]
[1,]    0    0    0    0
[2,]    0    0    0    0
[3,]    0    0    0    0
# display the dimension of matrix
dim(A0) 
[1] 3 4
# gives the number of rows of given matrix
nrow(A0) 
[1] 3
typeof(A0) 
[1] "double"
str(A0)
 num [1:3, 1:4] 0 0 0 0 0 0 0 0 0 0 ...

### Creating a matrix from a sequence of numbers

Matrix in R can also be created using sequence of numbers using : operator. By default, elements of sequence can be filled in a matrix column-wise.

Below R command create a matrix from sequence of numbers 1 to 9 with 3 rows and 3 columns.

B <- matrix(1:9, nrow = 3, ncol = 3)
B
     [,1] [,2] [,3]
[1,]    1    4    7
[2,]    2    5    8
[3,]    3    6    9

Elements of sequence can also be filled in a matrix row-wise by specifying byrow=T.

C <- matrix(1:9, nrow = 3, byrow = T)
C
     [,1] [,2] [,3]
[1,]    1    2    3
[2,]    4    5    6
[3,]    7    8    9

Number of elements should be in multiples of number of rows or columns, otherwise R display warning message.

If the number of elements are not multiple of number of rows or number of columns, then R display warning message and create a matrix by recycling the elements.

D <- matrix(1:9, nrow = 4)
Warning in matrix(1:9, nrow = 4): data length [9] is not a sub-multiple or
multiple of the number of rows [4]
D
     [,1] [,2] [,3]
[1,]    1    5    9
[2,]    2    6    1
[3,]    3    7    2
[4,]    4    8    3

The first command create a matrix from the sequence of numbers 1:9 with 4 rows and fill the elements column-wise and recycle the remaining elements form the start of the sequence.

### Creating a matrix from vector

Another way to create a matrix in R is to convert a vector into a matrix by specifying number of rows nrow or number of column ncol. By default the elements from the vector can be filled column-wise.

Suppose we have a vector v1=c(1,2,3,4,5,6). The second command create a matrix from vector v1 with number of rows equals 3 and store the matrix in A1.

v1 <- c(1, 2, 3, 4, 5, 6)
A1 <- matrix(v1, nrow = 3)
A1
     [,1] [,2]
[1,]    1    4
[2,]    2    5
[3,]    3    6
# Display the structure
str(A1) 
 num [1:3, 1:2] 1 2 3 4 5 6

If we specify byrow=T, then the matrix can be created from a vector v1 by filling the elements from the vector row-wise.

Below R code create a matrix B1 with number of rows 3 from vector v1 but this time it fill the elements by row as we have used the option byrow=T.

B1 <- matrix(v1, nrow = 3, byrow = T)
B1
     [,1] [,2]
[1,]    1    2
[2,]    3    4
[3,]    5    6
# Display number of rows
nrow(B1)   
[1] 3
# Display storage mode
typeof(B1)  
[1] "double"

### Creating a Matrix from vector using a dimension attributes

A matrix in R can also be constructed from a vector by setting the dimension attribute of given vector. Assigning a dimension to a vector treats it as a matrix of specified dimension.

Below R code create a vector da1.

# define a vector
da1 <- c(1, 2, 3, 4, 5, 6)
da1
[1] 1 2 3 4 5 6
class(da1)
[1] "numeric"

As da1 is a vector, the class(da1) display the result as numeric.

# set dimension of vector da1
# as 2 rows and 3 columns
dim(da1) <- c(2, 3)
da1
     [,1] [,2] [,3]
[1,]    1    3    5
[2,]    2    4    6

In the R code above, we have set the dimension of a vector da1 as c(2,3). R convert the vector da1 to a matrix of dimension 2 by 3. We can check this by using class() function.

class(da1)
[1] "matrix" "array" 

### Creating a matrix with dimension names

Rows and columns of a matrix can be labeled using dimnames() command.

C1 <- matrix(c(52,56,81,46,143,143,37,34), nrow = 2,
dimnames = list(c("Female","Male"),
c("Black", "Blond","Brown","Red")))
C1
       Black Blond Brown Red
Female    52    81   143  37
Male      56    46   143  34
# display dimension
dim(C1)
[1] 2 4
# Display dimension names
dimnames(C1)
[[1]]
[1] "Female" "Male"

[[2]]
[1] "Black" "Blond" "Brown" "Red"  

Row names and column names of a matrix can also be assigned using rownames() and colnames() function.

Let us create a matrix C2 from sequence of numbers 11 to 16 with number of rows equals 2.

C2 <- matrix(11:16, nrow = 2)
C2
     [,1] [,2] [,3]
[1,]   11   13   15
[2,]   12   14   16

Now assign the dimension names for C2 as follows:

rownames(C2) <- c("Row 1", "Row 2")
colnames(C2) <- c("Column 1", "Column 2", "Column 3")
dimnames(C2)
[[1]]
[1] "Row 1" "Row 2"

[[2]]
[1] "Column 1" "Column 2" "Column 3"

The first R code assign row names to matrix C2 and second R code assign column names to matrix C2. The third R code display the dimension names of matrix C2.

The row names and column names of matrix C2 can be retrieved using rownames(C2) and colnames(C2) command.

rownames(C2)
[1] "Row 1" "Row 2"
colnames(C2)
[1] "Column 1" "Column 2" "Column 3"

### Creating a matrix using cbind() and rbind() function

Matrices in R can also be created using cbind() (column-binding) or rbind() (row-binding) functions.

# combine columns
C3 <- cbind(c(1, 2), c(3, 4))
C3
     [,1] [,2]
[1,]    1    3
[2,]    2    4
# combine rows
C4 <- rbind(c(10, 13), c(23, 34))
C4
     [,1] [,2]
[1,]   10   13
[2,]   23   34

cbind() and rbind() can also be use to combine two or more matrices.

# combine two matrices using cbind
C5 <- cbind(C3, C4)
C5
     [,1] [,2] [,3] [,4]
[1,]    1    3   10   13
[2,]    2    4   23   34

To combine two or more matrices using cbind() function, the number of rows must be same in all the matrices.

# combine two matrices using rbind
C6 <- rbind(C3, C4)
C6
     [,1] [,2]
[1,]    1    3
[2,]    2    4
[3,]   10   13
[4,]   23   34

To combine two or more matrices using rbind() function, the number of columns must be same in all the matrices.

## Accessing matrix elements in R

### Accessing Rows/Columns using index

Elements of a matrix can be accessed by specifying row number(s) and/or column number(s). Like

• mat1[i,] returns $i^{th}$ row,
• mat[,j] returns $j^{th}$ column and
• mat1[i,j] returns $(i,j)^{th}$ element of matrix.

Let us create a matrix mat1 from sequence of numbers from 1 to 9 with 3 rows and elements are filled by row.

mat1 <- matrix(1:9, nrow = 3, byrow = TRUE)
mat1
     [,1] [,2] [,3]
[1,]    1    2    3
[2,]    4    5    6
[3,]    7    8    9

Below R code returns only first row of matrix mat1.

# returns 1st row of matrix mat1
mat1[1, ]   
[1] 1 2 3

Below R code returns only third column of matrix mat1.

# returns 3rd column of matrix mat1
mat1[, 3]   
[1] 3 6 9

Above R code returns third column of matrix mat1 but it return a vector, even though the object is a matrix. To prevent this from happening we use drop=FALSE argument as follows:

# returns 3rd column of matrix mat1
mat1[, 3, drop = FALSE]   
     [,1]
[1,]    3
[2,]    6
[3,]    9

Below R code return the element from second row and third column of matrix mat1.

# returns value from mat1 in the
# 2nd row and 3rd column
mat1[2, 3]  
[1] 6

Below R code returns the first and second row and third column of matrix mat1.

# returns the elements from mat1
# in the first 2 rows ad 3rd column
mat1[1:2, 3] 
[1] 3 6

To extract non-adjacent rows or columns, use c() (combine) function.

Below R code returns first and third row of matrix mat1.

# returns the elements from mat1
# in the first and third row
mat1[c(1, 3), ] 
     [,1] [,2] [,3]
[1,]    1    2    3
[2,]    7    8    9

The same can also be achieved by negative indexing. Below R code display all the rows except second row of matrix mat1.

# returns the elements from all
# rows except 2nd row
mat1[-2, ] 
     [,1] [,2] [,3]
[1,]    1    2    3
[2,]    7    8    9

### Accessing by Row/Column names

Recall the matrix C1 defined above.

C1
       Black Blond Brown Red
Female    52    81   143  37
Male      56    46   143  34

Below R code return the elements of row with row name Male from matrix C1.

C1["Male", ]
Black Blond Brown   Red
56    46   143    34 

Below R code return the elements of column with column name Non-smoker from matrix C1.

C1[, "Blond"]
Female   Male
81     46 

We can combine index and the row or column name to extract the data from given matrix.

Below R code return the elements from first row and the columns with column names Blond and Red.

C1[1, c("Blond", "Red")]
Blond   Red
81    37 

## Matrix Operations in R

• Arithmetic operations on matrices are element-wise: +, -, *,/
• Matrix multiplication: %*%
• Transpose of a matrix : t()
• To solve the system of linear equations Ax = b: solve(A,b)
• To find the inverse of a matrix $A^{-1}$ : solve(A)
• To computes eigen values and eigen vectors: eigen()
• To compute determinant of a matrix: det(A)

### Multiplication by scalar

A <- matrix(c(1, 2, 3, 4), nrow = 2)

Below R code multiply every element of A by 3.

A * 3
     [,1] [,2]
[1,]    3    9
[2,]    6   12

Addition and subtraction of matrices in R is performed element-wise.

B <- matrix(c(4, 5, 5, 3), nrow = 2)
# Addition of two matrices A and B
A + B 
     [,1] [,2]
[1,]    5    8
[2,]    7    7

Element-wise addition is performed on two matrices A and B.

# Difference of matrix B from A
A - B   
     [,1] [,2]
[1,]   -3   -2
[2,]   -3    1

### Matrix multiplication

A*A gives a new matrix with element-wise multiplication, where as A%*% A gives actual matrix multiplication.

A
     [,1] [,2]
[1,]    1    3
[2,]    2    4

Using * operator, the code A*A returns a new matrix with element-wise multiplication.

# Element-wise multiplication
A * A   
     [,1] [,2]
[1,]    1    9
[2,]    4   16

The actual matrix multiplication can be obtained using %*% operator.

# Matrix Multiplication
A %*% A 
     [,1] [,2]
[1,]    7   15
[2,]   10   22

The operator * perform element-wise matrix multiplication whereas the operator %*% perform actual matrix multiplication.

### Transpose of a matrix

The function t(x) returns the transpose of object x, where object x may be a matrix or data frame.

First R Code display matrix A and second R code compute the transpose of matrix A.

A
     [,1] [,2]
[1,]    1    3
[2,]    2    4
# transpose of a matrix A
t(A)   
     [,1] [,2]
[1,]    1    2
[2,]    3    4

### Inverse of a matrix

The arithmetic operations on matrices are element-wise. Hence if we use A^(-1), R will calculate a new matrix as a reciprocal of each element of matrix A.

# Matrix of reciprocal of each elements
A ^ (-1)  
     [,1]      [,2]
[1,]  1.0 0.3333333
[2,]  0.5 0.2500000

Above R code returns a matrix as a reciprocal of each element of matrix A.

Inverse of a square matrix in R can be computed using solve() function.

# Inverse of a matrix A
solve(A)  
     [,1] [,2]
[1,]   -2  1.5
[2,]    1 -0.5

### Determinant of a matrix

The det(x) function returns the determinant of matrix x.

# determinant of Matrix A
det(A)   
[1] -2

### Eigen values and eigen vectors of a matrix

The function eigen(x) computes the eigenvalues and eigen-vectors of matrix x.

# eigen values and eigen vectors of A
eigen(A)  
eigen() decomposition
$values [1] 5.3722813 -0.3722813$vectors
[,1]       [,2]
[1,] -0.5657675 -0.9093767
[2,] -0.8245648  0.4159736

Only eigen values can be extracted using following R code:

# gives only eigen values of A
eigen(A)$values  [1] 5.3722813 -0.3722813 Only eigen vectors can be extracted using following R code: # gives only eigen vectors of A eigen(A)$vectors 
           [,1]       [,2]
[1,] -0.5657675 -0.9093767
[2,] -0.8245648  0.4159736

## Addition functions for matrices in R

Some additional functions on matrices in R are as follows:

Function Explanation
diag() Create diagonal matrix or return diagonal
rowSums() Sum of elements of rows
colSums() Sum of elements of columns
rowMeans() Mean of elements of rows
colMeans() Mean of elements of columns

### diag() Function

The diag() function is very useful function for matrices. If the argument of a diag() function is a vector, it creates a diagonal matrix with the principal diagonal as elements of vector.

## Create a vector
a <- c(1, 3, 7)
a
[1] 1 3 7
A <- diag(a)
A
     [,1] [,2] [,3]
[1,]    1    0    0
[2,]    0    3    0
[3,]    0    0    7

If the argument of a diag() function is a matrix, it will display the principal diagonal elements.

B <- matrix(1:9, nrow = 3)
B
     [,1] [,2] [,3]
[1,]    1    4    7
[2,]    2    5    8
[3,]    3    6    9
## returns principal diagonal of B
diag(B)
[1] 1 5 9

If the argument of diag() function is a scalar, (say diag(m)) it returns identity matrix of order $m\times m$.

Below R code, returns a identity matrix of order $3\times 3$.

diag(3)
     [,1] [,2] [,3]
[1,]    1    0    0
[2,]    0    1    0
[3,]    0    0    1

### rowSums() and rowMeans() function on Matrix

The sum and means for each row of a matrix can be computed using rowSums() and rowMeans() functions respectively.

# compute the sum for each row of B
rowSums(B)
[1] 12 15 18
# compute the means for each row of B
rowMeans(B)
[1] 4 5 6

### colSums() and colMeans() function on Matrix

The sum and means for each column of a matrix can be computed using colSums() and colMeans() functions respectively. Check apply() function in R for more functions.

# compute the sum for each column of B
colSums(B)
[1]  6 15 24
# compute the mean for each column of B
colMeans(B)
[1] 2 5 8

Note that to compute rowSums, colSums, rowMeans, colMeans and other functions, we can use apply() function in R.

## Endnote

In this tutorial you learned about what is matrix data structure in R, how to create matrix, how to access subset of matrix in R and how to perform various operations on matrices in R with some illustrations.

Hopefully you enjoyed learning this tutorial on matrix type data structure in R. Hope the content is more than sufficient to understand matrix in R.