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If A is an n×n square matrix and λ is an eigenvalue of A, then the union of the zero vector and the set of all eigenvectors corresponding to eigenvalues λ is a subspace of Rn known as the eigenspace of λ.

(Mechanical vibration theory defines eigenvectors and eigenvalues as being analogous to mode shapes and natural frequencies of resonance, respectively.)

[In this shear mapping of the Mona Lisa, the picture was deformed in such a way that its central vertical axis (red vector) has not changed direction, but the diagonal vector (blue) has changed direction. Hence the red vector is an eigenvector of the transformation and the blue vector is not. Since the red vector was neither stretched nor compressed, its eigenvalue is 1. All vectors with the same vertical direction—i.e., parallel to this vector—are also eigenvectors, with the same eigenvalue. Together with the zero-vector, they form the eigenspace for this eigenvalue.]

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