(i) One way to compute eigenvalues is to solve the characteristic equation, either algebraically or numerically. Computers can employ some special algorithms to find approximate values for eigenvalues.
(ii) For complex matrices up to and including size , it is possible to compute the Jordan canonical form algebraically. However, computing JCF by hand is a bore; see workshop exercise W2.6.
(iii) In any method, multiple eigenvalues are a technical challenge. MATLAB has a command for finding the JCF; see W2.6.
(iv) However, computers find it easy to check whether matrices are positive definite, so researchers have found clever ways to use linear matrix inequalities instead of computing eigenvalues, as in frame 132.