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.
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