Computation of eigencomponents

The second equation is called the characteristic equation of the matrix A. Once A is known, the vector ut is computed as the unit vector solution of the linear system. It is left to the reader to show that the eigenvalues of a 2 x 2 matrix A can be found as a solution of the equation 

These equations, which give an eigendirection and not the coordinates of an eigenvector, are equivalent. An eigenvector, usually with unit length, must be chosen along that eigendirection, so that 

For the same eigenvalue, the opposite vector in the same eigendirection is the solution —x, — y, also acceptable. The eigenvector associated with A2 = 1 is 

Efficient computer programs rarely use this method. The numerical algorithms used to search for eigenvalues are countless and difficult to implement without a dramatic loss of accuracy. Use of the best canned routines saves time and produces better results. 

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