# The Solution of Matrix Eigenvalue Problems

• Ferenc Szidarovszky
• Sidney Yakowitz
Part of the Mathematical Concepts and Methods in Science and Engineering book series (MCSENG, volume 14)

## Abstract

Let A be a square matrix of order n. Consider the equation
$$Ax = \,\lambda x,$$
(7.1)
where λ is a real or complex number and x is an n-tuple. For x = 0, equation (7.1) is satisfied by an arbitrary scalar λ. The values of λ for which there exists some nonzero vector x satisfying (7.1) are called eigenvalues and the corresponding vectors x are called eigenvectors. Equation (7.1) can be written as
$$(\left. {A\, - \,\lambda } \right|)x\, = 0,$$
(7.2)
which for fixed λ is a homogeneous system of linear equations. This observation implies that if somehow we can obtain the eigenvalue λ of a given matrix A, we can find a corresponding eigenvector by solving (7.2) by methods from Chapter 6. Equation (7.2) has a nontrivial solution if and only if
$$\det (\left. {A - \,\lambda } \right|)\, = \,\left| {\begin{array}{*{20}{c}} {a{}_{11} - \lambda }&{a{}_{21}}& \cdots &{a{}_{1n}} \\ {a{}_{21}}&{a{}_{22} - \lambda }& \cdots &{a{}_{2n}} \\ \vdots &{}&{}& \vdots \\ {a{}_{n1}}&{a{}_{n2}}& \cdots &{a{}_{nn} - \lambda } \end{array}} \right| = 0.$$

## Keywords

Characteristic Polynomial Rayleigh Quotient Jacobi Method Roundoff Error Hessenberg Matrix
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.