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An eigenvector of an 𝑛 × 𝑛 matrix 𝐴 is
a nonzero vector x such that 𝐴𝐱 = λ𝐱 for some
scalar λ. A scalar λ is called an eigenvalue of 𝐴 if
there is a nontrivial solution x of 𝐴𝐱 = λ𝐱; such an
x is called an eigenvector corresponding to λ.
▪ λ is an eigenvalue of an 𝑛 × 𝑛 matrix 𝐴 if and only
if the equation (𝐴 − λ𝐼)𝐱 = 0 (1)
has a nontrivial solution.
▪ The set of all solutions of (1) is called the
eigenspace of 𝐴 corresponding to λ.

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