Creating Matrices
You can create matrices with the matrix function. The first argument is a vector, and the nrow and ncol arguments specify the number of rows and columns, respectively. Note that, by default, the elements of the matrix are filled in by column.
A <- matrix(c(1, 5, 3, 0), nrow = 2, ncol = 2)
B <- matrix(c(1, 7, 2, 1, -3, -1), nrow = 2, ncol = 3)
C <- matrix(1:4, nrow = 2, ncol = 2)The identity matrices can be created using the diag function. This creates the 2 x 2 diagonal matrix.
diag(2)## [,1] [,2]
## [1,] 1 0
## [2,] 0 1However, the diag function is one of the more surpsing functions in R; read the “Details” section of its documentation. These all do different things:
diag(3)## [,1] [,2] [,3]
## [1,] 1 0 0
## [2,] 0 1 0
## [3,] 0 0 1diag(nrow = 3)## [,1] [,2] [,3]
## [1,] 1 0 0
## [2,] 0 1 0
## [3,] 0 0 1diag(c(1, 2, 3))## [,1] [,2] [,3]
## [1,] 1 0 0
## [2,] 0 2 0
## [3,] 0 0 3diag(matrix(1:9, nrow = 3, ncol = 3))## [1] 1 5 9You can also create matrices by using cbind to combine vectors by column,
a <- c(1, 2, 3)
b <- c(4, 5, 6)
cbind(a, b)## a b
## [1,] 1 4
## [2,] 2 5
## [3,] 3 6and rbind to combine vectors by row,
rbind(a, b)## [,1] [,2] [,3]
## a 1 2 3
## b 4 5 6Note that cbind and rbind can take arbitrary number of arguments,
c <- c(7, 8, 9)
rbind(a, b, c)## [,1] [,2] [,3]
## a 1 2 3
## b 4 5 6
## c 7 8 9Matrix Information
To find the dimensions of a matrix use dim, ncol, or nrow
dim(B)## [1] 2 3nrow(B)## [1] 2ncol(B)## [1] 3To extract an element from a matrix use brackets. This extracts the 1st row, 2nd column from A,
A[1, 2]## [1] 3This extracts the 2nd row, 1st column from A,
A[2, 1]## [1] 5If you leave column blank, it extracts the entire row,
A[1, ]## [1] 1 3If you leave row blank, it extracts the entire column,
A[ , 1]## [1] 1 5You can also extract multiple rows or columns,
B[1, 2:3]## [1] 2 -3Matrix Operations
The common operators +, -, *, / and ^ work elementwise. In particular, * is not matrix multiplication.
A + C## [,1] [,2]
## [1,] 2 6
## [2,] 7 4A + 2## [,1] [,2]
## [1,] 3 5
## [2,] 7 2A - C## [,1] [,2]
## [1,] 0 0
## [2,] 3 -4A - 2## [,1] [,2]
## [1,] -1 1
## [2,] 3 -2A * C## [,1] [,2]
## [1,] 1 9
## [2,] 10 0A * 2## [,1] [,2]
## [1,] 2 6
## [2,] 10 0A / C## [,1] [,2]
## [1,] 1.0 1
## [2,] 2.5 0A / 2## [,1] [,2]
## [1,] 0.5 1.5
## [2,] 2.5 0.0A ^ C## [,1] [,2]
## [1,] 1 27
## [2,] 25 0A ^ 2## [,1] [,2]
## [1,] 1 9
## [2,] 25 0If you try to do operations with matrices that do not have comptible dimensions, you will get the following error.
A + B## Error in A + B: non-conformable arraysTo transpose a matrix use the t function
t(A)## [,1] [,2]
## [1,] 1 5
## [2,] 3 0t(B)## [,1] [,2]
## [1,] 1 7
## [2,] 2 1
## [3,] -3 -1For matrix multiplication use the %*% operator
A %*% C## [,1] [,2]
## [1,] 7 15
## [2,] 5 15t(C) %*% A## [,1] [,2]
## [1,] 11 3
## [2,] 23 9A %*% B## [,1] [,2] [,3]
## [1,] 22 5 -6
## [2,] 5 10 -15You can multiply a matrix by a vector, but it will treat the vector as a column vector.
B %*% c(1, 2, 3)## [,1]
## [1,] -4
## [2,] 6c(1, 2) %*% B## [,1] [,2] [,3]
## [1,] 15 4 -5but not,
B %*% c(1, 2)## Error in B %*% c(1, 2): non-conformable argumentsc(1, 2, 3) %*% B## Error in c(1, 2, 3) %*% B: non-conformable argumentsAside: To find help for a special function, quote its name after ?. For example,
?"%*%"To invert a matrix, use the solve function. This will calculate \(A^{-1}\),
solve(A)## [,1] [,2]
## [1,] 0.0000000 0.20000000
## [2,] 0.3333333 -0.06666667You cannot invert non-square matrices
solve(B)## Error in solve.default(B): 'a' (2 x 3) must be squareNote that you should avoid using solve if at all possible. Inverting matrices is computationally expensive (about \(O(n^3)\)), and there are more efficient methods to invert matrices using knowledge of features of the matrix.