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Math Origins: Eigenvectors and Eigenvalues | Mathematical Association of America

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In most undergraduate linear algebra courses, eigenvalues (and their cousins, the eigenvectors) play a prominent role. Their most immediate application is in transformational geometry, but they also appear in quantum mechanics, geology, and acoustics. Those familiar with the subject will know that a linear transformation 𝑇 𝑇 of an 𝑛 𝑛 -dimensional vector space can be represented by an 𝑛×𝑛 𝑛 × 𝑛 matrix, say 𝑀, 𝑀 , and that a nonzero vector 𝑣 ⃗  𝑣 → is an eigenvector of 𝑀 𝑀 if there is a scalar 𝜆 𝜆 for which 𝑀 𝑣 ⃗  =𝜆 𝑣 ⃗  . 𝑀 𝑣 → = 𝜆 𝑣 → . Narrowing our focus to 𝑛 𝑛 -dimensional real space, we may take 𝐼 𝐼 to be the 𝑛×𝑛 𝑛 × 𝑛 identity matrix, in which case the equation becomes (𝜆𝐼−𝑀) 𝑣 ⃗  = 0 ⃗  . ( 𝜆 𝐼 − 𝑀 ) 𝑣 → = 0 → . This equation can hold for a nonzero vector 𝑣 ⃗  𝑣 → (our eigenvector) only when the determinant of 𝜆𝐼−𝑀 𝜆 𝐼 − 𝑀 is zero. This leads us to a characteristic polynomial, defined by det(𝜆𝐼−𝑀). det ( �

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