Matrices

Matrix multiplication

The operator %*% is used for matrix multiplication. An n by 1 or 1 by n matrix may of course be used as an n-vector if in the context such is appropriate. Conversely, vectors which occur in matrix multiplication expressions are automatically promoted either to row or column vectors, whichever is multiplicatively coherent, if possible, (although this is not always unambiguously possible, as we see later).

If, for example, A and B are square matrices of the same size, then 

> A * B

is the matrix of element by element products and 

> A %*% B

is the matrix product. If x is a vector, then 

> x %*% A %*% x

is a quadratic form.

The function crossprod() forms "crossproducts", meaning that crossprod(X, y) is the same as t(X) %*% y but the operation is more efficient. If the second argument to crossprod() is omitted it is taken to be the same as the first.

The meaning of diag() depends on its argument. diag(v), where v is a vector, gives a diagonal matrix with elements of the vector as the diagonal entries. On the other hand diag(M), where M is a matrix, gives the vector of main diagonal entries of M. This is the same convention as that used for diag() in MATLAB. Also, somewhat confusingly, if k is a single numeric value then diag(k) is the k by k identity matrix!