This document provides brief explanations of the matrix functions available in Base R and the matrix package.

1. R Matrix Math Quick Reference
Mark Niemann-Ross
2021-05-14

2. About
About This Document
This document provides brief explanations of the matrix functions available in Base
R and the matrix package.
RelatedMaterial
This Quick Reference is supplemental to courses on LinkedIn Learning. A quick
reference to plotting functions in R is here. A quick reference to clustering functions
in R is here. An index to all R functions covered at LinkedIn Learning is found here.
The latest version of this quick reference is found here. The source to this document
is found on github/mnr. This document is available as Free Software under the
terms of the Free Software Foundation’s GNU General Public License.
About Mark Niemann-Ross
Mark is an author for LinkedIn Learning and writes Science Fiction.

3. Quick Reference to Matrix Math Functions for the R
programming language
This Quick Reference is supplemental to courses on LinkedIn Learning.
The latest version of this quick reference is found here
A quick reference to plotting functions in base R is here
A quick reference to clustering functions in R is here
An index to all R functions covered at LinkedIn Learning is found here
The Matrix package is recommended for advanced operations.

4. Create a matrix
Instructional video about Matrix
matrix( c(1:9), nrow = 3)
## [,1] [,2] [,3]
## [1,] 1 4 7
## [2,] 2 5 8
## [3,] 3 6 9

5. Addition
A <- matrix( c(1:9), nrow = 3)
B <- matrix( c(11:19), nrow = 3)
A
## [,1] [,2] [,3]
## [1,] 1 4 7
## [2,] 2 5 8
## [3,] 3 6 9
B
## [,1] [,2] [,3]
## [1,] 11 14 17
## [2,] 12 15 18
## [3,] 13 16 19
A + B
## [,1] [,2] [,3]
## [1,] 12 18 24

6. ## [2,] 14 20 26
## [3,] 16 22 28

7. Subtraction
A - B
## [,1] [,2] [,3]
## [1,] -10 -10 -10
## [2,] -10 -10 -10
## [3,] -10 -10 -10
B - A
## [,1] [,2] [,3]
## [1,] 10 10 10
## [2,] 10 10 10
## [3,] 10 10 10

8. Multiplication - simple
A * B
## [,1] [,2] [,3]
## [1,] 11 56 119
## [2,] 24 75 144
## [3,] 39 96 171

9. Multiplication - “dot product”
Instructional video about %*%
A %*% B
## [,1] [,2] [,3]
## [1,] 150 186 222
## [2,] 186 231 276
## [3,] 222 276 330

10. Division
HA - just kidding. There is no matrix division. Instead, use the inverse.
𝐴𝑋 = 𝐵 is solved for X with 𝑋 = 𝐴−1
𝐵
…lots more on inverse below…

11. The determinant of a matrix
Instructional video on det() and determinant()
The determinant of $begin{pmatrix} a & b c & d end{pmatrix}$ is 𝑎𝑑 − 𝑏𝑐
sampleMatrix <- matrix(c(10,12,5,30), nrow = 2)
sampleMatrix
## [,1] [,2]
## [1,] 10 5
## [2,] 12 30
det(sampleMatrix)
## [1] 240
…which is equivalent to 𝑎𝑑 − 𝑏𝑐 … which is written as…
sampleMatrix[1,1]*sampleMatrix[2,2] - sampleMatrix[1,2] *
sampleMatrix[2,1]

12. ## [1] 240

13. determinant vs det
Instructional video on det() and determinant()
determinant() produces a list with $modulus and $sign
determinant(sampleMatrix)
## $modulus
## [1] 5.480639
## attr(,"logarithm")
## [1] TRUE
##
## $sign
## [1] 1
##
## attr(,"class")
## [1] "det"
determinant(sampleMatrix, logarithm = FALSE)

14. ## $modulus
## [1] 240
## attr(,"logarithm")
## [1] FALSE
##
## $sign
## [1] 1
##
## attr(,"class")
## [1] "det"

15. Zero Matrix
matrix(0, nrow = 3, ncol = 3)
## [,1] [,2] [,3]
## [1,] 0 0 0
## [2,] 0 0 0
## [3,] 0 0 0

16. Identity Matrix
Instructional video on diag()
I <- diag(3)
I
## [,1] [,2] [,3]
## [1,] 1 0 0
## [2,] 0 1 0
## [3,] 0 0 1

17. Diagonal Matrix
Instructional video on diag()
diag(5, nrow = 3)
## [,1] [,2] [,3]
## [1,] 5 0 0
## [2,] 0 5 0
## [3,] 0 0 5
myMatrix <- A
diag(myMatrix) <- 8
myMatrix
## [,1] [,2] [,3]
## [1,] 8 4 7
## [2,] 2 8 8
## [3,] 3 6 8

18. Upper and Lower Triangular
Instructional video on lower.tri() and upper.tri()
upper.tri(A)
## [,1] [,2] [,3]
## [1,] FALSE TRUE TRUE
## [2,] FALSE FALSE TRUE
## [3,] FALSE FALSE FALSE
upper.tri(A, diag = TRUE)
## [,1] [,2] [,3]
## [1,] TRUE TRUE TRUE
## [2,] FALSE TRUE TRUE
## [3,] FALSE FALSE TRUE
lower.tri(A)
## [,1] [,2] [,3]
## [1,] FALSE FALSE FALSE

19. ## [2,] TRUE FALSE FALSE
## [3,] TRUE TRUE FALSE
myMatrix <- A
myMatrix[upper.tri(myMatrix)] <- NA
myMatrix
## [,1] [,2] [,3]
## [1,] 1 NA NA
## [2,] 2 5 NA
## [3,] 3 6 9

20. Matrix Comparison
There are two ways to compare matrices. First, create two matrices for example
comparison.
partiallyA <- A
partiallyA[upper.tri(partiallyA)] <- 1 # upper triangle becomes
different
simple matrice comparison, using equality
A == partiallyA # returns logical value for each element
## [,1] [,2] [,3]
## [1,] TRUE FALSE FALSE
## [2,] TRUE TRUE FALSE
## [3,] TRUE TRUE TRUE
object comparisonfor exact equality
identical(partiallyA, A) # returns logical value for entire matrice
comparison

21. ## [1] FALSE
identical(A,A) # compare two identical objects
## [1] TRUE

22. Matrix transposition
Instructional video on t()
NonSymmetricMatrix <- matrix(c(1:10), nrow = 5)
NonSymmetricMatrix # show what it looks like
## [,1] [,2]
## [1,] 1 6
## [2,] 2 7
## [3,] 3 8
## [4,] 4 9
## [5,] 5 10
t(NonSymmetricMatrix) # t() for transpose...show what transpose looks
like
## [,1] [,2] [,3] [,4] [,5]
## [1,] 1 2 3 4 5
## [2,] 6 7 8 9 10

23. Distance between points
Options to choose how distance is calculated, what type of matrix is produced. See
the instructional video for details.
mymatrix <- matrix( c(1:6), nrow = 3)
mymatrix
## [,1] [,2]
## [1,] 1 4
## [2,] 2 5
## [3,] 3 6
dist(mymatrix, method = "euclidean" )
## 1 2
## 2 1.414214
## 3 2.828427 1.414214

24. Build a symmetric matrix
There is a package for matrix tools. Here’s how to do it with baseR
A + t(A)
## [,1] [,2] [,3]
## [1,] 2 6 10
## [2,] 6 10 14
## [3,] 10 14 18

25. Build a skew-symmetrix matrix
A - t(A)
## [,1] [,2] [,3]
## [1,] 0 2 4
## [2,] -2 0 2
## [3,] -4 -2 0

26. Test for symmetric matrix
Instructional video on isSymmetric()
isSymmetric(A) # not symmetric
## [1] FALSE
isSymmetric(A + t(A)) # symmetric
## [1] TRUE
isSymmetric(A - t(A)) # skew symmetric
## [1] FALSE

27. Inner product of two vectors
Simple dot-product …aka 𝑣𝑒𝑐1𝑇
𝑣𝑒𝑐2
vec1 <- c(1:3)
vec2 <- c(1:3)
vec1 # what do these look like?
## [1] 1 2 3
vec2
## [1] 1 2 3
t(vec1) %*% vec2 # inner product
## [,1]
## [1,] 14

28. Outer product of two vectors
Instructional video on outer() and %o%
Transpose the second vector … 𝑣𝑒𝑐1𝑣𝑒𝑐2𝑇
These three versions produce the same result…
vec1 %*% t(vec2) # the actual formula
## [,1] [,2] [,3]
## [1,] 1 2 3
## [2,] 2 4 6
## [3,] 3 6 9
outer(vec1, vec2) # the R function
## [,1] [,2] [,3]
## [1,] 1 2 3
## [2,] 2 4 6
## [3,] 3 6 9

29. vec1 %o% vec2 # %o% is a wrapper for outer()
## [,1] [,2] [,3]
## [1,] 1 2 3
## [2,] 2 4 6
## [3,] 3 6 9

30. Outer product of two matrices
First…a reminder of the contents of matrix A and B
A
## [,1] [,2] [,3]
## [1,] 1 4 7
## [2,] 2 5 8
## [3,] 3 6 9
B
## [,1] [,2] [,3]
## [1,] 11 14 17
## [2,] 12 15 18
## [3,] 13 16 19
Then…here’s the outer product. This multiplies the first matrix by individual values
from the second matrix.
Think of this as…

31. A * B[1,1]
A * B[2,1]
…and so on
outer(A,B) # ... A %o% B will produce the same result
## , , 1, 1
##
## [,1] [,2] [,3]
## [1,] 11 44 77
## [2,] 22 55 88
## [3,] 33 66 99
##
## , , 2, 1
##
## [,1] [,2] [,3]
## [1,] 12 48 84
## [2,] 24 60 96
## [3,] 36 72 108
##
## , , 3, 1

32. ##
## [,1] [,2] [,3]
## [1,] 13 52 91
## [2,] 26 65 104
## [3,] 39 78 117
##
## , , 1, 2
##
## [,1] [,2] [,3]
## [1,] 14 56 98
## [2,] 28 70 112
## [3,] 42 84 126
##
## , , 2, 2
##
## [,1] [,2] [,3]
## [1,] 15 60 105
## [2,] 30 75 120
## [3,] 45 90 135
##
## , , 3, 2
##

33. ## [,1] [,2] [,3]
## [1,] 16 64 112
## [2,] 32 80 128
## [3,] 48 96 144
##
## , , 1, 3
##
## [,1] [,2] [,3]
## [1,] 17 68 119
## [2,] 34 85 136
## [3,] 51 102 153
##
## , , 2, 3
##
## [,1] [,2] [,3]
## [1,] 18 72 126
## [2,] 36 90 144
## [3,] 54 108 162
##
## , , 3, 3
##
## [,1] [,2] [,3]

34. ## [1,] 19 76 133
## [2,] 38 95 152
## [3,] 57 114 171

35. Solve a system of equations
Instructional video about solve()
As an example, start with this system of equations:
$$ 2x_1 - 3x_2 - 1x_3 = 2 1x_1 + 2x_2 + 3x_3 = 15 5x_1 + 1x_2 - 1x_3 = 4 $$
…when converted to a matrix…
$$begin{pmatrix} 2 & -3 & -1 1 & 2 & 3 5 & 1 & -1 end{pmatrix}
begin{pmatrix} x_1 x_2 x_3 end{pmatrix} = begin{pmatrix} 2 15
4 end{pmatrix}$$
…then, to solve with R…
coef_A <- matrix(c(2,1,5,-3,2,1,-1,3,-1), nrow = 3)
RHS_system <- matrix(c(2,15,4), nrow = 3)
solve(coef_A, RHS_system)

36. ## [,1]
## [1,] 2
## [2,] -1
## [3,] 5

37. Inverse matrix
Instructional video about solving for an inverse matrix
𝐴𝐴−
1 = 𝐼 … -1 is the inverse of a matrix. I is the identity matrix
solve(partiallyA) # solve() finds the inverse of a matrix
## [,1] [,2] [,3]
## [1,] 1.8571429 -0.1428571 -0.19047619
## [2,] -0.7142857 0.2857143 0.04761905
## [3,] -0.1428571 -0.1428571 0.14285714
Note: Some matrices return and error of Lapack routine dgesv: system is
exactly singular: U[3,3] = 0
Also note: solve() can be used in other ways. Refer to documentation.

38. Permutations
n <- 3 # for an n x n matrix
factorial(n) # provides the number of permutations
## [1] 6
BaseR doesn’t have a function to generate all possible permutations, but there are
packages that will do this.

39. backsolve
Instructional video about forward and backward solve for a matrix
Backsolve solves a triangular system of linear equations. This can come from a
gaussian elimination (for example).
Start with… begin{align} -3x_1 + 2x_2 - x_3 &= -1 -2x_2 + 5x_3 &= -9 -2x_3 &= 2
end{align}
# r holds a matrix of coefficients for the system to be solved
r <- matrix(c(-3, 2, -1,
0, -2, 5,
0, 0, -2),
nrow = 3, byrow = TRUE)
# x is a column matrix with the solutions
x <- matrix(c(-1, -9, 2), ncol = 1)
# backsolve produces the values of x.
backsolve(r, x)

40. ## [,1]
## [1,] 2
## [2,] 2
## [3,] -1
Backsolve produces a column matrix where 𝑚𝑎𝑡𝑟𝑖𝑥(𝑐(𝑥3,𝑥2,𝑥1),𝑛𝑐𝑜𝑙 = 1)

41. forwardsolve
Instructional video about forward and backward solve for a matrix
forwardsolve uses the lower triangular matrix. (backsolve() uses the upper
triangular matrix.) For example…
begin{align} 2 &= -2x_3 -9 &= 5x_3 -2x_2 -1 &= -x_3+ 2x_2 -3x_1 end{align}
# r holds a matrix of coefficients for the system to be solved
l <- matrix(c(-2, 0, 0,
5, -2, 0,
-1, 2, -3),
nrow = 3, byrow = TRUE)
# x is a column matrix with the solutions
x <- matrix(c(2, -9, -1), ncol = 1)
# forwardsolve produces the values of x.
forwardsolve(l, x)

42. ## [,1]
## [1,] -1
## [2,] 2
## [3,] 2

43. Singular Value Decomposition
Instructional video about SVD and QR decomposition
svd(A)
## $d
## [1] 1.684810e+01 1.068370e+00 5.543107e-16
##
## $u
## [,1] [,2] [,3]
## [1,] -0.4796712 0.77669099 0.4082483
## [2,] -0.5723678 0.07568647 -0.8164966
## [3,] -0.6650644 -0.62531805 0.4082483
##
## $v
## [,1] [,2] [,3]
## [1,] -0.2148372 -0.8872307 0.4082483
## [2,] -0.5205874 -0.2496440 -0.8164966
## [3,] -0.8263375 0.3879428 0.4082483

44. QR Decomposition
Instructional video about SVD and QR decomposition
Useful for solving 𝐴𝑥 = 𝑏 (𝐴 being a matrix, 𝑏 being a vector)
qr(A)
## $qr
## [,1] [,2] [,3]
## [1,] -3.7416574 -8.552360 -1.336306e+01
## [2,] 0.5345225 1.963961 3.927922e+00
## [3,] 0.8017837 0.988693 1.776357e-15
##
## $rank
## [1] 2
##
## $qraux
## [1] 1.267261e+00 1.149954e+00 1.776357e-15
##
## $pivot

45. ## [1] 1 2 3
##
## attr(,"class")
## [1] "qr"

46. crossproduct
Instructional video about crossproduct()
Not to be confused with a dot product (which is done with %*%). Regarding when
to use crossprod() vs t(A) %*% B … one may be faster than another based on the
matrix being sparse. But this will only make a difference on massive matrices.
crossprod(A,B)
## [,1] [,2] [,3]
## [1,] 74 92 110
## [2,] 182 227 272
## [3,] 290 362 434
…which is equivalent to…
t(A) %*% B
## [,1] [,2] [,3]
## [1,] 74 92 110

47. ## [2,] 182 227 272
## [3,] 290 362 434

48. tcrossproduct
tcrossprod(A,B)
## [,1] [,2] [,3]
## [1,] 186 198 210
## [2,] 228 243 258
## [3,] 270 288 306
…which is equivalent to…
A %*% t(B)
## [,1] [,2] [,3]
## [1,] 186 198 210
## [2,] 228 243 258
## [3,] 270 288 306

49. eigenvalues, eigenvectors
Instructional video about Eigenvalues and Eigenvectors()
eigen() returns both eigen values and eigen vectors
Here’s a matrix…
A
## [,1] [,2] [,3]
## [1,] 1 4 7
## [2,] 2 5 8
## [3,] 3 6 9
Here’s the eigen value and vector
eigen(A)
## eigen() decomposition
## $values
## [1] 1.611684e+01 -1.116844e+00 -5.700691e-16

50. ##
## $vectors
## [,1] [,2] [,3]
## [1,] -0.4645473 -0.8829060 0.4082483
## [2,] -0.5707955 -0.2395204 -0.8164966
## [3,] -0.6770438 0.4038651 0.4082483

51. Choleski Decomposition
symmetricalMatrix <- matrix(c(20,12,5,
12,15,2,
5,2,25), nrow = 3)
isSymmetric(symmetricalMatrix)
## [1] TRUE
chol(symmetricalMatrix)
## [,1] [,2] [,3]
## [1,] 4.472136 2.683282 1.1180340
## [2,] 0.000000 2.792848 -0.3580574
## [3,] 0.000000 0.000000 4.8602258

52. finally - look at package::matrix
#install.packages("Matrix")
library(Matrix)