Chapter 6 Matrices
A matrix is a table of data organized into rows and columns. \[\begin{equation} \mathbf{A} = \begin{bmatrix} a_{11} & a_{12} & \ldots & a_{1p} \\ a_{21} & a_{22} & \ldots & a_{2p} \\ \vdots & \vdots & \ddots & \vdots \\ a_{n1} & a_{12} & \ldots & a_{np} \end{bmatrix} \nonumber \end{equation}\]
The matrix \(\mathbf{A}\) has \(n\) rows and \(p\) columns.
The matrix element \(a_{ij}\) represents the number in the \(i^{th}\) row and \(j^{th}\) column.
R allows the elements of a matrix to be any type.
As with vectors, the elements of a matrix in R must all have the same type.
6.1 Creating matrices in R
The most common way to create a matrix is with the matrix function.
The form of the matrix function is
- x is a vector that you want to convert into a matrix.
nrowis the number of rows you want the matrix to have.ncolis the number of columns you want the matrix to have.xcould be a single number (a vector of length one) in which case the returned matrix will have all the same entries.
- If we wanted to create a matrix named
Awith 2 rows and 3 columns that contains only zeros, we could use the following code
## [,1] [,2] [,3]
## [1,] 0 0 0
## [2,] 0 0 0To convert a vector into a matrix
Convert the vector (1, 2, 3, 4, 5, 6) into a matrix with 2 rows and 3 columns
## [,1] [,2] [,3]
## [1,] 1 3 5
## [2,] 2 4 6- Notice that matrix function fills in the matrix entries by columns (i.e, it fills in the numbers from
xin the first column, then moves to the second, etc.)- This is referred to as filling in the entries “by column”
## [,1] [,2] [,3]
## [1,] 1 3 5
## [2,] 2 4 6- You can instead fill in the matrix entries by rows by using the keyword argument
byrow = TRUE
## [,1] [,2] [,3]
## [1,] 1 2 3
## [2,] 4 5 6- It is often useful to create a matrix which has all the same columns (or rows):
## [,1] [,2] [,3]
## [1,] 1 1 1
## [2,] 2 2 2## [,1] [,2] [,3]
## [1,] 1 3 2
## [2,] 2 1 3## [,1] [,2] [,3]
## [1,] 1 2 3
## [2,] 1 2 36.2 Accessing different features of a matrix
6.2.1 Accessing elements of a matrix
To access the \((i,j)\) element of a matrix
A, use the syntaxA[i,j]As an example of this, let’s create a \(2 \times 3\) matrix
Aand look at the \((2,1)\) element ofA:
## [,1] [,2] [,3]
## [1,] 1 2 3
## [2,] 4 5 6## [1] 4You can access different subsets of a matrix by using vectors of indeces rather than single numbers in the indexing bracket.
For example, if you want to look at elements \((2, 1)\) and \((2,2)\) of
A, you can use the following syntax:
## [1] 4 5- Similarly, if you want to access all elements of
Ain either the first two rows or first two columns ofA, you can use the following syntax:
## [,1] [,2]
## [1,] 1 2
## [2,] 4 5- To access the entire kth column of
A, use the syntaxA[,k].- To access the entire kth row of
A, use the syntaxA[k,]
- To access the entire kth row of
## [1] 2 5- Using nrow() or ncol() let you see the number of rows or columns of a matrix.
## [1] 2## [1] 3- dim will return both the number of rows and columns.
## [1] 2 3While using the
A[i,j]syntax is a more typical way of accessing specific elements of a matrix, you can also access the elements of a matrix as you would with a vector.For a matrix, the indexing increases by column
- For example, if
Ahas 3 rows: A[1]=A[1,1],A[2]=A[2,1],A[3]=A[3,1],A[4]=A[1,2], …
- For example, if
For example if we access the “4th” and “5th” elements of a \(2 x 3\) matrix
## [,1] [,2] [,3]
## [1,] 1 2 3
## [2,] 4 5 6## [1] 5 3- I don’t really use this way of accessing matrix elements much though,
in some cases, it is more convenient (e.g.,
A[ which(A > 3) ])
- You can also access elements of a matrix like a vector using logical subsetting
## [1] 2 4 6## [1] 4 5 66.2.2 Diagonals of Matrices
The diagonals of a matrix are the elements of the matrix where the row index equals the column index.
In a diagonal matrix, all non-diagonal elements must be zero.
To create a diagonal matrix in R, use the
diagfunction.- You must provide the vector of diagonal elements as the input to
diag
- You must provide the vector of diagonal elements as the input to
## [,1] [,2] [,3] [,4]
## [1,] 1 0 0 0
## [2,] 0 2 0 0
## [3,] 0 0 3 0
## [4,] 0 0 0 4- If you use
diag(A)where the input argumentAis a matrix, this will return the diagonal elements ofA.
## [,1] [,2] [,3]
## [1,] 1 4 7
## [2,] 2 5 8
## [3,] 3 6 9## [1] 1 5 9Working with off-diagonal elements.
The function
upper.tri(A)returns a logical matrix of the same dimension asAwithTRUEvalues on the “upper diagonal” part of the matrix.- Thus, you can use
upper.trito update the upper diagonal parts of a matrix.
- Thus, you can use
## [,1] [,2] [,3]
## [1,] 1 0 0
## [2,] 2 5 0
## [3,] 3 6 9- Similarly,
lower.tri(A)allows you to look at the subset of entries on the “lower diagonal” part of a matrix.
## [,1] [,2] [,3]
## [1,] 10 0 0
## [2,] 11 13 0
## [3,] 12 14 156.2.3 Naming rows or columns
- You can give a name to each row with the rownames function.
- The collection of row names of a matrix should be a vector.
## [,1] [,2] [,3]
## a 1 3 5
## b 2 4 6- You can actually access rows of a matrix by name:
## [1] 1 3 5## [1] 1 3 5- Similarly, you can access columns of a matrix by name:
## c1 c2 c3
## a 1 3 5
## b 2 4 6## c2 c3
## 4 6## c2 c3
## 4 66.2.4 Combining rows or columns
- You can “join” matrices in “vertical” or “horizontal” ways
by using the functions
cbind(column bind)rbind(row bind)
## [,1] [,2] [,3] [,4] [,5] [,6]
## [1,] 1 3 5 1 9 25
## [2,] 2 4 6 4 16 36## [,1] [,2] [,3]
## [1,] 1 3 5
## [2,] 2 4 6
## [3,] 1 9 25
## [4,] 4 16 366.3 Applying functions to matrices
You can directly use matrices as input arguments with many of the widely-used built-in R functions.
Functions like sum or mean will take the sum (or average) of all the elements of a matrix:
## [1] 21## [1] 3.5## [1] 1.870829- You can take the transpose of a matrix
Ausingt(A).- You can convert a matrix
Ainto a vector usingas.vector(A)or simplyc(A).
- You can convert a matrix
## [,1] [,2] [,3]
## [1,] 1 3 5
## [2,] 2 4 6## [,1] [,2]
## [1,] 1 2
## [2,] 3 4
## [3,] 5 6## [1] 1 2 3 4 5 6## [1] 1 3 5 2 4 66.3.1 Mathematical operations with matrices
- Using
+,-,*,/with two matrices performs element-wise addition, subtraction, multiplication, and division.
## [,1] [,2] [,3]
## [1,] 2 4 6
## [2,] 8 10 12## [,1] [,2] [,3]
## [1,] 0 0 0
## [2,] 0 0 0## [,1] [,2] [,3]
## [1,] 1 4 9
## [2,] 16 25 36Note that
A*Bdoes not perform true matrix multiplication of matricesAandB.Matrix multiplication is done with
%*%
## [,1] [,2]
## [1,] 14 32
## [2,] 32 77## [,1] [,2] [,3]
## [1,] 17 22 27
## [2,] 22 29 36
## [3,] 27 36 456.4 Matrix Functionals
6.4.1 Functions applied to each row/column
- We can take the sum or mean of each row/column of a matrix by using either
rowSums,rowMeans,colSums, orcolMeans.
## [,1] [,2] [,3]
## [1,] 1 3 5
## [2,] 2 4 6## [1] 9 12## [1] 3 7 11## [1] 3 4- While
rowSumsworks perfectly fine, let’s try writing our own rowSums() function:
myRowSums <-function(x){
ret <- rep(NA,nrow(x)) ## make an empty vector
for(i in 1:nrow(x)) {
ret[i] <-sum(x[i,]) ## take the sum of i-th row
}
return (ret)
}
rowSums(A)## [1] 9 12## [1] 9 12The implementation by the function
myRowSumsworks fine.However, it is a bit cumbersome to write such a function whenever we want to compute column-wise/row-wise summary statistics.
It would be nice if there were a quick & general way to do the following:
- For each row or column of a matrix …
- Evaluate a function using the row/column vector as input
- Return the collection of results as a numeric vector.
6.4.2 apply(): dimension-wise aggregation
## [1] 9 12## [1] 9 12## [1] 3 7 11## [1] 3 7 11First parameter of apply: the input matrix.
Second parameter of apply: the dimension over which to aggregate
- 1:rows, 2:columns
Third parameter of apply: function to evaluate for each row/column
## [,1] [,2] [,3]
## [1,] 1 3 5
## [2,] 2 4 6## [1] 9 12## [1] 3 7 11- Examples of using apply:
## [1] 1.5 3.5 5.5## [1] 1.5 3.5 5.5## [1] 2 4 66.4.3 Using apply with your own function
Suppose that instead of simply taking the row-wise or column-wise means we want to do the following:
For each row/column:
- Calculate the squared sum of each row/column.
- Return the vector of squared sums.
How can we do this using apply()?
A big advantage of the apply function is the flexibility it gives you.
You can use your own functions to compute row-wise/column-wise summary measures for certain measures that may not be implemented in base R.
## define a function named sqsum
sqsum <- function(x) {
return( sum(x*x) ) ## very simple implementation
}
apply(A, 1, sqsum) ## run sqsum() funciton## [1] 35 56You can actually write the function definition inside of the apply function.
This can be convenient when using straightforward functions that can be written on a single line:
## [1] 35 56- Or, we can write the above apply statement in an even simpler form:
## You can omit return() and {} especially
## for simple function definitions
apply(A, 1, function(x) sum(x*x) ) ## Even simpler!## [1] 35 56- apply() can also return a matrix
## [,1] [,2]
## [1,] 1 2
## [2,] 5 6## [,1] [,2] [,3]
## [1,] 1 3 5
## [2,] 2 4 66.5 Exercises
- Write a function that has two input arguments A and B
- A and B are both matrices with the same dimension (same number of rows and columns).
- The function returns a matrix D.
- The
[i,j]element of D should equalA[i,j]ifA[i,j] < B[i,j]and equalB[i,j]ifA[i,j] >= B[i,j]. - You can use either a nested for loop or logical indexing.
- Suppose we define the matrix
Xas
What will be the value of apply(X, 2, sum)[2]?
- Write a function called
PairwiseMedianDiffsthat has two numeric matricesAandBas the input arguments. These two input matrices are assumed to have the same dimensions. This function should return a \(p \times p\) matrix (where \(p\) is the number of columns inA). The[i,j]element of the returned matrix should equal the median of the \(i^{th}\) column ofAminus the median of the \(j^{th}\) column ofB.