The Big R-Book. Philippe J. S. De Brouwer
nrow = 4, byrow = TRUE) M ## [,1] [,2] [,3] ## [1,] 10 11 12 ## [2,] 13 14 15 ## [3,] 16 17 18 ## [4,] 19 20 21 # Access one element: M[1,2] ## [1] 11 # The second row: M[2,] ## [1] 13 14 15 # The second column: M[,2] ## [1] 11 14 17 20 # Row 1 and 3 only: M[c(1, 3),] ## [,1] [,2] [,3] ## [1,] 10 11 12 ## [2,] 16 17 18 # Row 2 to 3 with column 3 to 1 M[2:3, 3:1] ## [,1] [,2] [,3] ## [1,] 15 14 13 ## [2,] 18 17 16
4.3.4.4 Matrix Arithmetic
Matrix arithmetic allows both base operators and specific matrix operations. The base operators operate always element per element.
Basic arithmetic on matrices works element by element:
M1 <- matrix( c(10:21), nrow = 4, byrow = TRUE) M2 <- matrix(c(:11), nrow = 4, byrow = TRUE) M1 + M2 ## [,1] [,2] [,3] ## [1,] 10 12 14 ## [2,] 16 18 20 ## [3,] 22 24 26 ## [4,] 28 30 32 M1 * M2 ## [,1] [,2] [,3] ## [1,] 0 11 24 ## [2,] 39 56 75 ## [3,] 96 119 144 ## [4,] 171 200 231 M1 / M2 ## [,1] [,2] [,3] ## [1,] Inf 11.000000 6.000000 ## [2,] 4.333333 3.500000 3.000000 ## [3,] 2.666667 2.428571 2.250000 ## [4,] 2.111111 2.000000 1.909091
Write a function for the dot-product for matrices. Add also some security checks. Finally, compare your results with the “%*%-operator.”
dot-product
The dot-product is pre-defined via the %*% opeartor. Note that the function t() creates the transposed vector or matrix.
# Example of the dot-product: a <- c(1:3) a %*% a ## [,1] ## [1,] 14 a %*% t(a) ## [,1] [,2] [,3] ## [1,] 1 2 3 ## [2,] 2 4 6 ## [3,] 3 6 9 t(a) %*% a ## [,1] ## [1,] 14 # Define A: A <- matrix(:8, nrow = 3, byrow = TRUE) # Test products: A %*% a ## [,1] ## [1,] 8 ## [2,] 26 ## [3,] 44 A %*% t(a) # this is bound to fail! ## Error in A %*% t(a): non-conformable arguments A %*% A ## [,1] [,2] [,3] ## [1,] 15 18 21 ## [2,] 42 54 66 ## [3,] 69 90 111
There are also other operations possible on matrices. For example the quotient works as follows:
A %/% A ## [,1] [,2] [,3] ## [1,] NA 1 1 ## [2,] 1 1 1 ## [3,] 1 1 1
Note that matrices will accept both normal operators and specific matrix operators.
# Note the difference between the normal product: A * A ## [,1] [,2] [,3] ## [1,] 0 1 4 ## [2,] 9 16 25 ## [3,] 36 49 64 # and the matrix product %*%: A %*% A ## [,1] [,2] [,3] ## [1,] 15 18 21 ## [2,] 42 54 66 ## [3,] 69 90 111 # However, there is -of course- only one sum: A + A ## [,1] [,2] [,3] ## [1,] 0 2 4 ## [2,] 6 8 10 ## [3,] 12 14 16 # Note that the quotients yield almost the same: A %/% A ## [,1] [,2] [,3] ## [1,] NA 1 1 ## [2,] 1 1 1 ## [3,] 1 1 1 A / A ## [,1] [,2] [,3] ## [1,] NaN 1 1 ## [2,] 1 1 1 ## [3,] 1 1 1
The same hold for quotient and other operations.
Note that while exp(A), for example, is well defined for a matrix as the sum of the series:
R will resort to calculating the exp() element by element!
Using the same matrix A as in the aforementioned code:
# This is the matrix A: A ## [,1] [,2] [,3] ## [1,] 0 1 2 ## [2,] 3 4 5 ## [3,] 6 7 8 # The exponential of A: exp(A) ## [,1] [,2] [,3] ## [1,] 1.00000 2.718282 7.389056 ## [2,] 20.08554 54.598150 148.413159 ## [3,] 403.42879 1096.633158 2980.957987
The same holds for all other functions of base R:
# The natural logarithm log(A) ## [,1] [,2] [,3] ## [1,] -Inf 0.000000 0.6931472 ## [2,] 1.098612 1.386294 1.6094379 ## [3,] 1.791759 1.945910 2.0794415 sin(A) ## [,1] [,2] [,3] ## [1,] 0.0000000 0.8414710 0.9092974 ## [2,] 0.1411200 -0.7568025 -0.9589243 ## [3,] -0.2794155 0.6569866 0.9893582
Note also that some operations will collapse the matrix to another (simpler) data type.
# Collapse to a vectore: colSums(A) ## [1] 9 12 15 rowSums(A) ## [1] 3 12 21 # Some functions aggregate the whole matrix to one scalar: mean(A) ## [1] 4 min(A) ## [1] 0
We already saw the function t() to transpose a matrix. There are a few others available in base R. For example, the function diag() diagonal matrix that is a subset of the matrix, det() caluclates the determinant, etc. The function solve() will solve the equation A %.% x = b, but when A is missing, it will assume the idenity vector and return the inverse of A.
diag()
solve()
M <- matrix(c(1,1,4,1,2,3,3,2,1), 3, 3) M ## [,1] [,2] [,3] ## [1,] 1 1 3 ## [2,] 1 2 2 ## [3,] 4 3 1 # The diagonal of M: diag(M) ## [1] 1 2 1 # Inverse: solve(M) ## [,1] [,2] [,3] ## [1,] 0.3333333 -0.66666667 0.33333333 ## [2,] -0.5833333 0.91666667 -0.08333333 ## [3,] 0.4166667 -0.08333333 -0.08333333 # Determinant: det(M) ## [1] -12 # The QR composition: QR_M <- qr(M) QR_M$rank ## [1] 3 # Number of rows and columns: nrow(M) ## [1] 3 ncol(M) ## [1] 3 # Sums of rows and columns: colSums(M) ## [1] 6 6 6 srowSums(M) ## [1] 5 5 8 # Means of rows, columns, and matrix: colMeans(M) ## [1] 2 2 2 rowMeans(M) ## [1] 1.666667 1.666667 2.666667 mean(M) ## [1] 2 # Horizontal and vertical concatenation: rbind(M, M) ## [,1] [,2] [,3] ## [1,] 1 1 3 ## [2,] 1 2 2 ## [3,] 4 3 1 ## [4,] 1 1 3 ## [5,] 1 2 2 ## [6,] 4 3 1 cbind(M, M) ## [,1] [,2] [,3] [,4] [,5] [,6] ## [1,] 1 1 3 1 1 3 ## [2,] 1 2 2 1 2 2 ## [3,] 4 3 1 4 3 1
4.3.5 Arrays
Matrices are very useful, however there will be times that data has more dimensions than just two. R has a solutions with the base-type “array.” Unlike matrices that have always two dimensions, arrays can be of any number of dimensions. However, the requirement that all elements are of the same data-type is also valid for arrays.
array
Note that words like “array” are used as keywords in many computer languages, and that it is important to understand exactly how it is implemented in the language that you want to use. In this section we will introduce you to the practical aspects of working with arrays.
4.3.5.1 Creating and Accessing Arrays
Arrays