Given a vector , the symbol denotes its Euclidean norm, i.e. . We denote the th element of as . The th element of the canonical basis of is represented by . To ease the readability, column vectors are also expressed as .
Consider the matrix , then denotes the th column of , the th row of , and the th element of . Moreover, denotes the transpose of . Given a square matrix , . To simplify the notation, we express diagonal matrices as , where are the diagonal elements of the matrix.
The symbol denotes the identity matrix. The symbol refers to the th eigenvalue of . In particular, , denote the largest and the smallest eigenvalue of , respectively. A matrix is said to be positive semidefinite if and for all , and is said to be positive definite if the inequality is strict, i.e. for all . is negative (semi)definite if is positive (semi)definite. For a positive definite matrix and a vector , we denote the weighted Euclidean norm as . The notation used for constant matrices is directly extended to the nonconstant case.
Unless something different is stated, all the functions treated in this book are assumed to be smooth. Moreover, the symbol is reserved to express time, where we assume . Then, given a function that depends on time, the symbol denotes the differentiation with respect to time of , i.e. where . The and norms of signals are denoted and
