In this chapter we will discuss, in some detail, some iterative methods for finding single eigenvalue–eigenvector pairs (*eigenpairs* is a common term) of a given real matrix *A*; we will also give an overview of more powerful and general methods that are commonly used to find *all* the eigenpairs of a given real *A.* As in Chapter 7, our discussion here will depend a fair amount on MATLAB, although we will look at some algorithms in detail.

The algebraic eigenvalue problem is as follows: Given a matrix , find a nonzero vector *x* ^{n} and the scalar λ such that

Note that this says that the vector *Ax* is parallel to *x*, with λ being an amplification factor, or *gain.* Note also that the above implies that

showing (by Theorem 7.1) that *A* − λ*I* is a singular matrix. Hence, det(*A* − λ*I*) = 0; it is easy to show that this determinant is a polynomial (of degree *n*) in λ, known as the *characteristic polynomial* of *A, p*(λ), so ...

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