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The dimension of this vector space is the number of pixels. The eigenvectors of the covariance matrix associated with a large set of normalized pictures of faces are called eigenfaces; this is an example of principal component analysis. They are very useful for expressing any face image as a linear combination of some of them.
Similarly, if M is a hypersurface in a Riemannian manifold N, then the principal curvatures are the eigenvalues of its second-fundamental form. If k 1, ..., k n are the n principal curvatures at a point p ∈ M and X 1, ..., X n are corresponding orthonormal eigenvectors (principal directions), then the sectional curvature of M at p is given by
The k-th principal component of a data vector x (i) can therefore be given as a score t k(i) = x (i) ⋅ w (k) in the transformed coordinates, or as the corresponding vector in the space of the original variables, {x (i) ⋅ w (k)} w (k), where w (k) is the kth eigenvector of X T X. The full principal components decomposition of X can therefore ...
Fold mountains form in areas of thrust tectonics, such as where two tectonic plates move towards each other at convergent plate boundary.When plates and the continents riding on them collide or undergo subduction (that is – ride one over another), the accumulated layers of rock may crumple and fold like a tablecloth that is pushed across a table, particularly if there is a mechanically weak ...
Notation: The index j represents the jth eigenvalue or eigenvector. The index i represents the ith component of an eigenvector. Both i and j go from 1 to n, where the matrix is size n x n. Eigenvectors are normalized. The eigenvalues are ordered in descending order.
The eigenvalues are real. The eigenvectors of A −1 are the same as the eigenvectors of A. Eigenvectors are only defined up to a multiplicative constant. That is, if Av = λv then cv is also an eigenvector for any scalar c ≠ 0. In particular, −v and e iθ v (for any θ) are also eigenvectors.
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In general, an eigenvector of a linear operator D defined on some vector space is a nonzero vector in the domain of D that, when D acts upon it, is simply scaled by some scalar value called an eigenvalue. In the special case where D is defined on a function space, the eigenvectors are referred to as eigenfunctions.