CHAPTER SIX Eigensystems

6.1 EIGENVALUES AND EIGENVECTORS

The purpose of computing is insight, not numbers.

—RICHARD W. HAMMING (1915–1998)

Numbers are intellectual witnesses that belong only to mankind.

—HONORE DE BALZAC (1799–1850)

Introduction

Sometimes, the effect of multiplying a vector by a matrix is to produce simply a scalar multiple of that vector. For example,

 

We often want to know to what extent this phenomenon occurs for a given matrix. That is, we ask for all instances of the equation Ax = λx, where A is given but we know neither the vector x nor the scalar λ. Because there are two unknowns on the right multiplied together, this is ...

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