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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.
However the rank of the covariance matrix is limited by the number of training examples: if there are N training examples, there will be at most N − 1 eigenvectors with non-zero eigenvalues. If the number of training examples is smaller than the dimensionality of the images, the principal components can be computed more easily as follows.
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 ...
Given an n × n square matrix A of real or complex numbers, an eigenvalue λ and its associated generalized eigenvector v are a pair obeying the relation [1] =,where v is a nonzero n × 1 column vector, I is the n × n identity matrix, k is a positive integer, and both λ and v are allowed to be complex even when A is real.l When k = 1, the vector is called simply an eigenvector, and the pair ...
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.
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
Dubai featured several times in the list, with Dubai-Riyadh the busiest route in the Middle East. - Asia-Pacific Images Studio/E+/Getty Images. When it comes to domestic routes, the top three ...
Also (in the maximum theorem) subsequent eigenvalues and eigenvectors are found by induction and orthogonal to each other; therefore, = with , =, <. The Courant minimax principle, as well as the maximum principle, can be visualized by imagining that if || x || = 1 is a hypersphere then the matrix A deforms that hypersphere into an ellipsoid .