<P> The orthogonality properties of the eigenvectors allows decoupling of the differential equations so that the system can be represented as linear summation of the eigenvectors . The eigenvalue problem of complex structures is often solved using finite element analysis, but neatly generalize the solution to scalar - valued vibration problems . </P> <P> In image processing, processed images of faces can be seen as vectors whose components are the brightnesses of each pixel . 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 . In the facial recognition branch of biometrics, eigenfaces provide a means of applying data compression to faces for identification purposes . Research related to eigen vision systems determining hand gestures has also been made . </P> <P> Similar to this concept, eigenvoices represent the general direction of variability in human pronunciations of a particular utterance, such as a word in a language . Based on a linear combination of such eigenvoices, a new voice pronunciation of the word can be constructed . These concepts have been found useful in automatic speech recognition systems for speaker adaptation . </P> <P> In mechanics, the eigenvectors of the moment of inertia tensor define the principal axes of a rigid body . The tensor of moment of inertia is a key quantity required to determine the rotation of a rigid body around its center of mass . </P>

Eigen values and eigen vectors of a square matrix