Appendix E. Matrix Algebra in R

Many of the functions described in this book operate on matrices. The manipulation of matrices is built deeply into the R language. Table E.1 describes operators and functions that are particularly important for solving linear algebra problems. In the following table, A and B are matrices, x and b are vectors, and k is a scalar.

Table E.1. R functions and operators for matrix algebra

Operator or Function	Description
+ - * / Λ	Element-wise addition, subtraction, multiplication, division, and exponentiation, respectively.
A %*% B	Matrix multiplication.
A %o% B	Outer product. AB'.
cbind(A, B, ...)	Combine matrices or vectors horizontally.
chol(A)	Choleski factorization of A. If R <- chol(A), then chol(A) contains the upper triangular factor, such that R’R = A.
colMeans(A)	Returns a vector containing the column means of A.
crossprod(A)	A’A.
crossprod(A,B)	A’B.
colSums(A)	Returns a vector containing the column sums of A.
diag(A)	Returns a vector containing the elements of the principal diagonal.
diag(x)	Creates a diagonal matrix with the elements of x in the principal diagonal.
diag(k)	If k is a scalar, this creates a k x k identity matrix.
eigen(A)	Eigenvalues and eigenvectors of A. If y <- eigen(A), then y$val are the eigenvalues of A and y$vec are the eigenvectors of A.
ginv(A)	Moore-Penrose Generalized Inverse of A. (Requires the MASS package).
qr(A)	QR decomposition of A. If y <- qr(A), then y$qr has an upper triangle containing the decomposition and a lowertriangle that contains information on the decomposition, y$rank is the rank of A, y$qraux is a vector containing additional information on Q, and y$pivot contains information on the pivoting strategy used.
rbind(A, B, ...)	Combines matrices or vectors vertically.
rowMeans(A)	Returns a vector containing the row means of A.
rowSums(A)	Returns a vector containing the row sums of A.
solve(A)	Inverse of A where A is a square matrix.
solve(A, b)	Solves for vector x in the equation b = Ax.
svd(A)	Single value decomposition of A. If y <- svd(A), then y$d is a vector containing the singular values of A, y$u is a matrix with columns containing the left singular vectors of A, and y$v is a matrix with columns containing the right singular vectors of A.
t(A)	Transpose of A.