Vector and matrix algebra

Packages for this section

This is (almost) all base R! We only need this for one thing later:

library(tidyverse)

Vector addition

Adding vectors:

u <- c(2, 3, 6, 5, 7)
v <- c(1, 8, 3, 2, 0)
u + v

[1]  3 11  9  7  7

Elementwise addition. (Linear algebra: vector addition.)

Adding a number to a vector

Define a vector, then “add 2” to it:

[1] 2 3 6 5 7

k <- 2
u + k

[1] 4 5 8 7 9

adds 2 to each element of u.

Scalar multiplication

As per linear algebra:

[1] 2

[1] 2 3 6 5 7

k * u

[1]  4  6 12 10 14

Each element of vector multiplied by 2.

“Vector multiplication”

What about this?

[1] 2 3 6 5 7

[1] 1 8 3 2 0

u * v

[1]  2 24 18 10  0

Each element of u multiplied by corresponding element of v. Could be called elementwise multiplication.

(Don’t confuse with “outer” or “vector” product from linear algebra, or indeed “inner” or “scalar” multiplication, for which the answer is a number.)

Combining different-length vectors

No error here (you get a warning). What happens?

[1] 2 3 6 5 7

w <- c(1, 2)
u + w

[1] 3 5 7 7 8

Add 1 to first element of u, add 2 to second.
Go back to beginning of w to find something to add: add 1 to 3rd element of u, 2 to 4th element, 1 to 5th.

How R does this

Keep re-using shorter vector until reach length of longer one.
“Recycling”.
If the longer vector’s length not a multiple of the shorter vector’s length, get a warning (probably not what you want).
Same idea is used when multiplying a vector by a number: the number keeps getting recycled.

Matrices

Create matrix like this:

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

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

First: stuff to make matrix from, then how many rows and columns.
R goes down columns by default. To go along rows instead:

(B <- matrix(5:8, nrow = 2, ncol = 2, byrow = TRUE))

     [,1] [,2]
[1,]    5    6
[2,]    7    8

One of nrow and ncol enough, since R knows how many things in the matrix.

Adding matrices

What happens if you add two matrices?

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

     [,1] [,2]
[1,]    5    6
[2,]    7    8

A + B

     [,1] [,2]
[1,]    6    9
[2,]    9   12

Adding matrices

Nothing surprising here. This is matrix addition as we and linear algebra know it.

Multiplying matrices

Now, what happens here?

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

     [,1] [,2]
[1,]    5    6
[2,]    7    8

A * B

     [,1] [,2]
[1,]    5   18
[2,]   14   32

Multiplying matrices?

Not matrix multiplication (as per linear algebra).
Elementwise multiplication. Also called Hadamard product of A and B.

Legit matrix multiplication

Like this:

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

     [,1] [,2]
[1,]    5    6
[2,]    7    8

A %*% B

     [,1] [,2]
[1,]   26   30
[2,]   38   44

Reading matrix from file

The usual:

my_url <- "http://ritsokiguess.site/datafiles/m.txt"
M <- read_delim(my_url, " ", col_names = FALSE )
M

class(M)

[1] "spec_tbl_df" "tbl_df"      "tbl"         "data.frame"

but…

except that M is not an R matrix, and thus this doesn’t work:

v <- c(1, 3)
M %*% v

Error in M %*% v: requires numeric/complex matrix/vector arguments

Making a genuine matrix

Do this first:

M <- as.matrix(M)
M

     X1 X2
[1,] 10  9
[2,]  8  7
[3,]  6  5

[1] 1 3

and then all is good:

M %*% v

     [,1]
[1,]   37
[2,]   29
[3,]   21

Linear algebra stuff

To solve system of equations \(Ax = w\) for \(x\):

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

[1] 1 2

solve(A, w)

[1] 1 0

Matrix inverse

To find the inverse of A:

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

solve(A)

     [,1] [,2]
[1,]   -2  1.5
[2,]    1 -0.5

Checking

Matrix inverse:

A %*% solve(A)

     [,1] [,2]
[1,]    1    0
[2,]    0    1

System of equations:

A %*% solve(A, w)

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

[1] 1 2

Thus, matrix inverse and equation solution are correct.

Inner product

Vectors in R are column vectors, so just do the matrix multiplication (t() is transpose):

a <- c(1, 2, 3)
b <- c(4, 5, 6)
t(a) %*% b

     [,1]
[1,]   32

Note that the answer is actually a 1 × 1 matrix.
Or as the sum of the elementwise multiplication:

sum(a * b)

[1] 32

Accessing parts of vector

use square brackets and a number to get elements of a vector

[1] 4 5 6

b[2]

[1] 5

Accessing parts of matrix

use a row and column index to get an element of a matrix

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

A[2,1]

[1] 2

leave the row or column index empty to get whole row or column, eg.

A[1,]

[1] 1 3

Eigenvalues and eigenvectors 1/2

For a matrix \(A\), these are scalars \(\lambda\) and vectors \(v\) that solve

\[ A v = \lambda v \]

In R, eigen gets these:

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

e <- eigen(A)

Eigenvalues and eigenvectors 2/2

eigen() decomposition
$values
[1]  5.3722813 -0.3722813

$vectors
           [,1]       [,2]
[1,] -0.5657675 -0.9093767
[2,] -0.8245648  0.4159736

Eigenvalues/vectors correct?

\(\lambda_1 v_1\): (scalar) multiply first eigenvalue by first eigenvector (in column)

e$values[1] * e$vectors[,1]

[1] -3.039462 -4.429794

\(A v_1\): (matrix) multiply matrix by first eigenvector (in column)

A %*% e$vectors[,1]

          [,1]
[1,] -3.039462
[2,] -4.429794

These are (correctly) equal.
The second one goes the same way.

A statistical application of eigenvalues

A negative correlation:

d <- tribble(
  ~x,  ~y,
  10,  20,
  11,  18,
  12,  17,
  13,  14,
  14,  13
)
v <- cor(d)
v

           x          y
x  1.0000000 -0.9878783
y -0.9878783  1.0000000

cor gives the correlation matrix between each pair of variables (correlation between x and y is \(-0.988\))

Eigenanalysis of correlation matrix

eigen(v)

eigen() decomposition
$values
[1] 1.98787834 0.01212166

$vectors
           [,1]       [,2]
[1,] -0.7071068 -0.7071068
[2,]  0.7071068 -0.7071068

first eigenvalue much bigger than second (second one near zero)
two variables, but data nearly one-dimensional
opposite signs in first eigenvector indicate that the one dimension is:
- x small and y large at one end,
- x large and y small at the other.