For various applications, we need to quantify errors and distances—i.e., we need to measure the size of a vector and a matrix. We shall find that it is not only the Euclidian measure of a vector that is appropriate in practice. Accordingly, we introduce vector and matrix norms, which are real valued functions of vectors and matrices, respectively. It is shown how to calculate matrix norms associated with certain vector norms and apply them in the estimation of eigenvalues. We shall find that properties of positive definite matrices play a crucial role in the calculation of the norm associated with the Euclidian vector norm. We show also that matrix norms are useful when estimating inherent errors—i.e., errors caused by errors in given data—in solutions of systems of linear algebraic equations. Finally, it is shown how to estimate the errors in eigenvalue computations by a posteriori estimates and how to compute sequences of approximations of them by matrix power methods. The following definitions are introduced in this appendix.