Linear Algebra for Data Science with Python by Shea John M.;

Author:Shea, John M.;
Language: eng
Format: epub
Publisher: CRC Press LLC
Published: 2025-09-24T00:00:00+00:00

for some real constant λ.

Here, eigen is the German word for own, meaning that for a given matrix, such vectors are special characteristics of that matrix. Any value λ that satisfies this equation for some eigenvector is called an eigenvalue of :

Definition

eigenvalue

Given a square matrix , a constant λ is an eigenvalue of if there exists a non-zero vector such that

!

Important!

There may be multiple vectors that satisfy (3.2), so we add a subscript i to distinguish them. Each eigenvector has an associated eigenvalue λi to satisfy (3.2). It is best to think of them as an eigenvector-eigenvalue pair, .

Suppose that are an eigenvector-eigenvalue pair of a matrix . Consider a vector , where c is a constant. Then

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