# eigen - EIGENVALUES AND EIGENVECTORS 1 Definition They are...

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EIGENVALUES AND EIGENVECTORS 1. Definition They are defined in terms of each other. Let A be an n × n matrix. A vector v = 0 is an eigenvector of A with eigenvalue λ if the equation Av = λv is satisfied. Note that eigenvectors are not uniquely defined: If v is an eigenvector then any scalar multiple of v is also an eigenvector. In fact, if u and v are two eigenvectors for the same eigenvalue λ , then any linear combination a · u + b · v is also an eigenvector with eigenvalue λ . 2. Applications of eigenvectors You might wonder why anyone would be interested in eigenvectors and eigenval- ues. Eigenvectors are vectors toward which the matrix A behaves like a scalar namely the scalar λ (the corresponding eigenvalue). If we could find a basis consisting of eigenvectors, A would become a diagonal matrix in this basis: its action on basis-vectors would only be to multiply each of them by a scalar — which is what a diagonal matrix does. Diagonal matrices are interesting because the are easy to work with — they behave like scalars when you add or multiply them.

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