It is possible for a real or complex matrix to have all real eigenvalues without being Hermitian. For example, a real triangular matrix has its eigenvalues along its diagonal, but in general is not symmetric.Īny problem of numeric calculation can be viewed as the evaluation of some function f for some input x. The condition number κ( f, x) of the problem is the ratio of the relative error in the function's output to the relative error in the input, and varies with both the function and the input. The condition number describes how error grows during the calculation. Its base-10 logarithm tells how many fewer digits of accuracy exist in the result than existed in the input. The condition number is a best-case scenario. It reflects the instability built into the problem, regardless of how it is solved. No algorithm can ever produce more accurate results than indicated by the condition number, except by chance. However, a poorly designed algorithm may produce significantly worse results. For example, as mentioned below, the problem of finding eigenvalues for normal matrices is always well-conditioned. However, the problem of finding the roots of a polynomial can be very ill-conditioned. Thus eigenvalue algorithms that work by finding the roots of the characteristic polynomial can be ill-conditioned even when the problem is not.įor the problem of solving the linear equation A v = b where A is invertible, the condition number κ( A −1, b) is given by || A|| op|| A −1|| op, where || || op is the operator norm subordinate to the normal Euclidean norm on C n. Since this number is independent of b and is the same for A and A −1, it is usually just called the condition number κ( A) of the matrix A. This value κ( A) is also the absolute value of the ratio of the largest eigenvalue of A to its smallest. For general matrices, the operator norm is often difficult to calculate. For this reason, other matrix norms are commonly used to estimate the condition number.įor the eigenvalue problem, Bauer and Fike proved that if λ is an eigenvalue for a diagonalizable n × n matrix A with eigenvector matrix V, then the absolute error in calculating λ is bounded by the product of κ( V) and the absolute error in A. As a result, the condition number for finding λ is κ( λ, A) = κ( V) = || V || op || V −1|| op. If A is normal, then V is unitary, and κ( λ, A) = 1. Thus the eigenvalue problem for all normal matrices is well-conditioned.
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