# 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
$$\det (\left. {A - \,\lambda } \right|)\, = \,\left| {\begin{array}{*{20}{c}} {a{}_{11} - \lambda }&{a{}_{21}}& \cdots &{a{}_{1n}} \\ {a{}_{21}}&{a{}_{22} - \lambda }& \cdots &{a{}_{2n}} \\ \vdots &{}&{}& \vdots \\ {a{}_{n1}}&{a{}_{n2}}& \cdots &{a{}_{nn} - \lambda } \end{array}} \right| = 0.$$

Assure Posite