Matrices

(Sections 7.1 - 7.3)

What R Was Designed For

R for Data Analysis

R was designed for statistics and data analysis. It supports data structures that make data analysis easier.

In this Chapter we learn about one such data structure: data frames.

Unlike the vectors you have met so far, data frames are two-dimensional.

So, as a warm-up, we look at another two-dimensional data structure in R: the matrix.

Introduction to Matrices

What a Matrix Is

Matrix

An atomic vector that has two additional attributes: a number of rows and a number of columns.

Creating a Matrix

Take a vector and give it those two extra attributes, using the matrix() function.:

Alternative

You could just specify one of the dimensions:

Fill in Across Rows

What if you want to “write in” the elements by row, instead of down columns?

Any Data Type Allowed

Matrices do not have to be numerical:

Spreading Out a Matrix

You can convert a matrix to a regular vector:

Naming Rows and Columns

You can give your matrix row and column-names!

rownames(numbersMat) <- letters[1:6]  # six rows
colnames(numbersMat) <- LETTERS[1:4]  # four columns
numbersMat

  A  B  C  D
a 1  7 13 19
b 2  8 14 20
c 3  9 15 21
d 4 10 16 22
e 5 11 17 23
f 6 12 18 24

Matrix Indexing

Set Up numbersMat Yourself

Please run this code:

Using Indices

You subset matrices with the [ operator:

The row and column numbers are called indices.

General Form

Suppose that vector1 and vector2 are numerical vectors. Then

numbersMat[vector1, vector2]

gives you a sub-matrix of the elements in rows and columns covered by combinations of values in the two vectors.

Example

The sub-matrix consisting of elements in

• the second or fourth rows, and
• the first, second or third columns

can be gotten like this:

Example

The sub-matrix consisting of elements in

• the second or fourth rows, and
• any of the four columns

can be gotten like this:

Example

The sub-matrix consisting of elements in

• any of the six rows and
• the first, second or third columns

can be gotten like this:

A (Slight) Exception

If you only want to pick from one row (or one column) then you don’t get a matrix back.

The result is a vector, not a matrix. (Usually that’s what you want.)

What If I WANT a Matrix Back?

Set the drop parameter to FALSE:

Caution: Watch Indices

Don’t ask for rows or columns that your matrix does not have:

Subsetting with Names …

… works just as with ordinary vectors:

Logical Subsetting …

… works just as with ordinary vectors:

Subsetting to Assign: I

numbersMat[2,3] <- 0
numbersMat

  A  B  C  D
a 1  7 13 19
b 2  8  0 20
c 3  9 15 21
d 4 10 16 22
e 5 11 17 23
f 6 12 18 24

Subsetting to Assign: II

You can assign to more than one place in a matrix:

numbersMat[2,] <- 0
numbersMat

  A  B  C  D
a 1  7 13 19
b 0  0  0  0
c 3  9 15 21
d 4 10 16 22
e 5 11 17 23
f 6 12 18 24

The 0 was recycled to make this happen!

Subsetting to Assign: III

You can assign different values to different places.

numbersMat[2,] <- c(100, 200, 300, 400)
numbersMat

    A   B   C   D
a   1   7  13  19
b 100 200 300 400
c   3   9  15  21
d   4  10  16  22
e   5  11  17  23
f   6  12  18  24

Operations on Matrices

Arithmetic Operations

Here are two 2-by-2 matrices:

mat1 <- matrix(rep(1, 4), nrow = 2)
mat2 <- matrix(rep(2, 4), nrow = 2)
mat1

     [,1] [,2]
[1,]    1    1
[2,]    1    1

mat2

     [,1] [,2]
[1,]    2    2
[2,]    2    2

Addition

Matrices add element-wise:

mat1 + mat2

     [,1] [,2]
[1,]    3    3
[2,]    3    3

Subtraction, division, etc. all work the element-wise as well.

Recycling Applies

Suppose we have:

mat <- matrix(1:4, nrow = 2)
mat

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

Multiply every element of mat by 2:

2 * mat

     [,1] [,2]
[1,]    2    6
[2,]    4    8

Recycling

This is the same as:

matrix(rep(2, 4), nrow = 2) * mat

Matrix Multiplication

* does not do the matrix multiplication that we do in Linear Algebra.

To learn how R does “real” matrix multiplication, study the textbook.

Logical Operations

Boolean operations apply to matrices element-wise. The result is a matrix of logical values. Consider the original matrix numbersMat:

numbersMat <- matrix(1:24, nrow = 6)

Determine which elements of numbersMat are odd:

numbersMat %% 2 == 1

      [,1]  [,2]  [,3]  [,4]
[1,]  TRUE  TRUE  TRUE  TRUE
[2,] FALSE FALSE FALSE FALSE
[3,]  TRUE  TRUE  TRUE  TRUE
[4,] FALSE FALSE FALSE FALSE
[5,]  TRUE  TRUE  TRUE  TRUE
[6,] FALSE FALSE FALSE FALSE

Logical Subsetting

We can select elements from a matrix using a Boolean operator:

numbersMat[numbersMat %% 2 == 1]

 [1]  1  3  5  7  9 11 13 15 17 19 21 23

The result is an ordinary, one-dimensional vector.