Principles and Procedures of Numerical Analysis pp 221-256 | Cite as

# The Solution of Matrix Eigenvalue Problems

Chapter

## Abstract

Let where which for fixed

**A**be a square matrix of order*n*. Consider the equation$$Ax = \,\lambda x,$$

(7.1)

*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)

*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.

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© Springer Science+Business Media New York 1978