1.1 Basic Notation

In this course we will refer frequently to matrices, vectors, and scalars. A matrix will be denoted by an upper case letter such as A, and its ( i, j)th element will be denoted by a _ij. If the matrix is given by an expression such as A + B, we will write ( A + B) _ij. In detailed algorithmic descriptions we will sometimes write A( i, j) or use the Matlab ^[1] [184] notation A(i : j,k : l) to denote the submatrix of A lying in rows i through j and columns k through l. A lower-case letter like x will denote a vector, and its ith element will be written x _i. Vectors will almost always be column vectors, which are the same as matrices with one column. Lower-case Greek letters (and occasionally lower-case letters) will denote scalars. will denote the set of real numbers; , the set of n-dimensional real vectors; and , the set of m-by- n real matrices. , and denote complex numbers, vectors, and matrices, respectively. Occasionally we will use the shorthand A ^{m n} to indicate that A is an m-by- n matrix. A ^T will denote the transpose of the matrix A: ( A ^T) _ij = a _ji