5 Matrices
Introduction
A matrix is a series of numbers arranged in rows and columns. We usually abbreviate matrices with a bolded uppercase letter:
\mathbf{A}_{j \times k} = \begin{bmatrix} a_{1,1} & a_{1,2} & \cdots & a_{1,k} \\ a_{2,1} & a_{2,2} & \cdots & a_{2,k} \\ \vdots & \vdots & \ddots & \vdots \\ a_{j,1} & a_{j,2} & \cdots & a_{j,k} \end{bmatrix}
For example,
\mathbf{A} = \begin{bmatrix} 1&2&3&4\\ 5&6&7&8\\ 9&10&11&12 \end{bmatrix}
To make a matrix in R, there are many possibilities. For the sake of simplicity, I will recommend using the matrix function with the byrow option set to TRUE.
The matrix function takes a vector and converts it to a matrix. By default, the matrix function assumes that the vector is a single column. For example,
matrix(1:4) [,1]
[1,] 1
[2,] 2
[3,] 3
[4,] 4By setting the nrow (number of rows) or ncol (number of columns), the matrix dimensions are set. For example,
matrix(1:4, ncol = 2) [,1] [,2]
[1,] 1 3
[2,] 2 4Note that the default is to fill in the matrix by columns. If you want it to fill by rows, set the byrow option to TRUE.
matrix(1:4, ncol = 2, byrow = TRUE) [,1] [,2]
[1,] 1 2
[2,] 3 4In general, creating matrices by row makes the code and the matrix look similar:
matrix(c(3, 5,
2, 3),
nrow = 2,
byrow = TRUE) [,1] [,2]
[1,] 3 5
[2,] 2 3To achieve the same matrix without the byrow option set to TRUE is possible, but the code will not be in the same order as the matrix:
matrix(c(3, 2,
5, 3),
nrow = 2) [,1] [,2]
[1,] 3 5
[2,] 2 35.1 Appending Matrices
\color{RoyalBlue}{\mathbf{A}}=\begin{bmatrix} \color{RoyalBlue}{1} & \color{RoyalBlue}{2} & \color{RoyalBlue}{3} & \color{RoyalBlue}{4} \\ \color{RoyalBlue}{5} & \color{RoyalBlue}{6} & \color{RoyalBlue}{7} & \color{RoyalBlue}{8} \\ \color{RoyalBlue}{9} & \color{RoyalBlue}{10} & \color{RoyalBlue}{11} & \color{RoyalBlue}{12} \end{bmatrix}, \color{Firebrick}{\mathbf{B}}=\begin{bmatrix} \color{Firebrick}{13} & \color{Firebrick}{14} & \color{Firebrick}{15} & \color{Firebrick}{16} \\ \color{Firebrick}{17} & \color{Firebrick}{18} & \color{Firebrick}{19} & \color{Firebrick}{20} \end{bmatrix}
The equation below means, “Make a new matrix called C by appending B to the bottom of A.”
\mathbf{C}=\begin{bmatrix} \color{RoyalBlue}{\mathbf{A}} \\ \color{FireBrick}{\mathbf{B}} \end{bmatrix}= \begin{bmatrix} \color{RoyalBlue}{1} & \color{RoyalBlue}{2} & \color{RoyalBlue}{3} & \color{RoyalBlue}{4} \\ \color{RoyalBlue}{5} & \color{RoyalBlue}{6} & \color{RoyalBlue}{7} & \color{RoyalBlue}{8} \\ \color{RoyalBlue}{9} & \color{RoyalBlue}{10} & \color{RoyalBlue}{11} & \color{RoyalBlue}{12}\\ \color{Firebrick}{13} & \color{Firebrick}{14} & \color{Firebrick}{15} & \color{Firebrick}{16} \\ \color{Firebrick}{17} & \color{Firebrick}{18} & \color{Firebrick}{19} & \color{Firebrick}{20} \end{bmatrix}
In R, adding new rows to a matrix is done with the rbind function (Think “row bind”). To append rows, matrices must have the same number of columns to be compatible. If the matrices are incompatible, no appending is possible.
A <- matrix(1:12, nrow = 3, byrow = TRUE)
B <- matrix(13:20, nrow = 2, byrow = TRUE)
C <- rbind(A,B)
C [,1] [,2] [,3] [,4]
[1,] 1 2 3 4
[2,] 5 6 7 8
[3,] 9 10 11 12
[4,] 13 14 15 16
[5,] 17 18 19 20The cbind function (Think “column bind”) works the same way but it appends columns to the right of a matrix. Matrices must have the same number of rows to be compatible.
A <- matrix(1:4, nrow = 2)
B <- matrix(5:8, nrow = 2)
C <- cbind(A,B)
C [,1] [,2] [,3] [,4]
[1,] 1 3 5 7
[2,] 2 4 6 85.2 Selecting parts of a matrix
Often we need to select smaller portions of a larger matrix. For example, suppose we have a 3 by 4 matrix A:
\mathbf{A} = \begin{bmatrix} 1&2&3&4\\ 5&6&7&8\\ 9&10&11&12 \end{bmatrix}
A <- matrix(1:12, nrow = 3, byrow = TRUE)
A [,1] [,2] [,3] [,4]
[1,] 1 2 3 4
[2,] 5 6 7 8
[3,] 9 10 11 125.2.1 Select a single element of a matrix
To select a single element of a matrix, specify the row and column in brackets after the matrix. For example, the element \mathbf{A}_{3,2} (i.e., the 3rd row and 2nd column of \mathbf{A}) is
A[3,2][1] 105.2.2 Select a matrix row
Leaving one of the slots in the bracket empty means that you want all of the elements in that row or column.
\mathbf{A}_{1\bullet} is the 1st row of \mathbf{A}.
A[1, ][1] 1 2 3 45.2.3 Select a matrix column
\mathbf{A}_{\bullet 3} is the 3rd column of \mathbf{A}.
A[, 3][1] 3 7 11By default, whenever a single row, column, or element is returned from a matrix, R drops the row and column dimensions. If you wish to preserve the result in matrix form, set drop to FALSE:
A[, 3, drop = FALSE] [,1]
[1,] 3
[2,] 7
[3,] 11Select column 1 of A.
Suggested Solution
A[, 1][1] 1 5 95.2.4 Select several columns or rows
A vector of integers will select whichever rows or columns you wish. Here are the 2nd and 3rd rows:
A[2:3, ] [,1] [,2] [,3] [,4]
[1,] 5 6 7 8
[2,] 9 10 11 12Here are the 1st and 4th columns:
A[, c(1, 4)] [,1] [,2]
[1,] 1 4
[2,] 5 8
[3,] 9 12Select columns 1 and 3 of A.
Suggested Solution
A[, c(1, 3)] [,1] [,2]
[1,] 1 3
[2,] 5 7
[3,] 9 115.2.5 Selecting with Boolean vectors
Here is the first two rows of \mathbf{A}:
A[c(TRUE,TRUE,FALSE),] [,1] [,2] [,3] [,4]
[1,] 1 2 3 4
[2,] 5 6 7 8This seems like a strange way to do this, but it is actually quite powerful. Any vector of TRUE and FALSE values can be used to select things. For example, select a column only if its first value is greater than 2:
# Create a vector that tests whether the
# first row elements are greater than 2
s <- A[1, ] > 2
# Select only the columns whose
# first value is greater than 2
A[, s] [,1] [,2]
[1,] 3 4
[2,] 7 8
[3,] 11 12Use a Boolean vector to select columns that begin with 2. The operator for “is equal to” is ==.
Suggested Solution
s <- A[1,] == 2
# When only one column or row is selected, R returns a vector
A[,s][1] 2 6 10# To force R to return a matrix, set drop = FALSE
A[,s, drop = FALSE] [,1]
[1,] 2
[2,] 6
[3,] 105.2.6 Selecting with name vectors
Suppose matrix A represents the number of Olympic medals a country has won in different gymnastics events. We can give row and column names to a matrix like so:
rownames(A) <- c("Gold","Silver","Bronze")
colnames(A) <- c("Vault","Uneven Bars","Balance Beam","Floor")
A Vault Uneven Bars Balance Beam Floor
Gold 1 2 3 4
Silver 5 6 7 8
Bronze 9 10 11 12Now we can select rows and columns by names. To select only the gold and bronze meal rows:
A[c("Gold","Bronze"),] Vault Uneven Bars Balance Beam Floor
Gold 1 2 3 4
Bronze 9 10 11 12Use the column names to select the Uneven Bars column
Suggested Solution
A[,"Uneven Bars"] Gold Silver Bronze
2 6 10 # or to force R to return a matrix
A[,"Uneven Bars", drop = FALSE] Uneven Bars
Gold 2
Silver 6
Bronze 10Select the intersection of Silver and Floor
Suggested Solution
A["Silver", "Floor"][1] 8# or
A["Silver", "Floor", drop = FALSE] Floor
Silver 85.3 Replace portions of a matrix
Any portion of a matrix can be replaced with new values. For example, this will replace the first row with zeros:
A[1, ] <- c(0, 0, 0, 0)
A Vault Uneven Bars Balance Beam Floor
Gold 0 0 0 0
Silver 5 6 7 8
Bronze 9 10 11 12This can be done by column or row name as well
A["Gold", ] <- c(0, 0, 0, 0)
A Vault Uneven Bars Balance Beam Floor
Gold 0 0 0 0
Silver 5 6 7 8
Bronze 9 10 11 12Replace the Vault column of \mathbf{A} with a vector of (10,20,30)
Suggested Solution
A[, "Vault"] <- c(10, 20, 30)
A Vault Uneven Bars Balance Beam Floor
Gold 10 0 0 0
Silver 20 6 7 8
Bronze 30 10 11 125.4 Transposing matrices
To transpose a matrix is to flip its rows into columns.
\mathbf{A}' is matrix \mathbf{A} transposed.
If
\mathbf{A}=\begin{bmatrix} 1&2&3\\ 4&5&6 \end{bmatrix}
Then
\mathbf{A}'=\begin{bmatrix} 1&4\\ 2&5\\ 3&6 \end{bmatrix}
5.4.1 Transposing in R
In R the t function transposes matrices.
A <- matrix(1:6, nrow = 2, byrow = TRUE)
A [,1] [,2] [,3]
[1,] 1 2 3
[2,] 4 5 6A_prime <- t(A)
A_prime [,1] [,2]
[1,] 1 4
[2,] 2 5
[3,] 3 6- Make a 2 \times 5 matrix
Aof even numbers up to 20. - Transpose
A, assigning it to a variable calledA_prime. - Replace the last row (row 5) of
A_primewith zeroes.
Suggested Solution
A <- matrix(seq(2, 20, 2), nrow = 2)
A [,1] [,2] [,3] [,4] [,5]
[1,] 2 6 10 14 18
[2,] 4 8 12 16 20A_prime <- t(A)
A_prime [,1] [,2]
[1,] 2 4
[2,] 6 8
[3,] 10 12
[4,] 14 16
[5,] 18 20A_prime[5, ] <- c(0, 0)
A_prime [,1] [,2]
[1,] 2 4
[2,] 6 8
[3,] 10 12
[4,] 14 16
[5,] 0 05.5 Square matrices
In a square matrix, the number of rows is equal to the number of columns. Some operations in matrix algebra require square matrices. To test whether a matrix is square, test whether the rows and columns of a matrix are equal:
# A is a 3 by 4 matrix, so it is not square
nrow(A) == ncol(A)[1] FALSE5.6 Symmetric matrices
A symmetric matrix is a square matrix that is equal to its transpose.
\mathbf{A}=\mathbf{A}'
This means that for all elements, a_{ij}=a_{ji}.
Here is an example of a symmetric matrix:
\begin{bmatrix} \color{green}a & \color{Firebrick}b & \color{RoyalBlue}c\\ \color{Firebrick}b & \color{gold}d & \color{DarkOrchid}e\\ \color{RoyalBlue}c & \color{DarkOrchid}e & \color{orange}f \end{bmatrix}
To test whether a matrix is symmetric:
isSymmetric(A)[1] FALSECorrelation matrices and covariance matrices are always symmetric.
5.7 Diagonal matrices
A diagonal matrix is a square matrix consisting of zeroes everywhere except the diagonal. For example,
\mathbf{A} = \begin{bmatrix} a & 0 & 0\\ 0 & b & 0\\ 0 & 0 & c \end{bmatrix}
5.7.1 Creating diagonal matrices
As will be seen, the diag function does different things depending on the type of input it receives. If its input is a vector, it will create a diagonal matrix. That is, the diag function creates a diagonal matrix \mathbf{A} from a vector \vec{a} like so:
# Make a vector
a <- 1:4
# Make a diagonal matrix
A <- diag(a)
A [,1] [,2] [,3] [,4]
[1,] 1 0 0 0
[2,] 0 2 0 0
[3,] 0 0 3 0
[4,] 0 0 0 45.7.2 Extracting the diagonal vector from a matrix
The diag function extracts a diagonal vector \vec{a} from a matrix \mathbf{A}.
diag(A)[1] 1 2 3 45.7.3 Replacing the diagonal of a matrix
The diag function can also be used to replace a diagonal of a matrix:
diag(A) <- 5:8
A [,1] [,2] [,3] [,4]
[1,] 5 0 0 0
[2,] 0 6 0 0
[3,] 0 0 7 0
[4,] 0 0 0 8