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Eigenvector (including recent related patents.)
EigenvectorTopics in Linear Algebra Vectors Vector spaces Linear span Linear transformation Linear independence Linear combination Basis Column space Row space Dual space Orthogonality Eigenvector Eigenvalue Least squares regressions Outer product Cross product Dot product Transpose Matrix decomposition In linear algebra, the eigenvectors (from the German eigen meaning "inherent, characteristic") of a linear operator are non-zero vectors which, when operated on by the operator, result in a scalar multiple of themselves. The scalar is then called the eigenvalue associated with the eigenvector. In applied mathematics and physics the eigenvectors of a matrix or a differential operator often have important physical significance. In classical mechanics the eigenvectors of the governing equations typically correspond to natural modes of vibration in a body, and the eigenvalues to their frequencies. In quantum mechanics, operators correspond to observable variables, eigenvectors are also called eigenstates, and the eigenvalues of an operator represent those values of the corresponding variable that have non-zero probability of occurring. Table of contents showTocToggle("show","hide") 1 Examples 2 Definition 3 Finding eigenvectors 4 The characteristic polynomial 5 Complex eigenvectors 6 Infinite-dimensional spaces 7 External links Examples Intuitively, for linear transformations of two-dimensional space R2, eigenvectors are thus:
and because p(x) = - (x - 2)(x - 1)(x + 1) we see that the eigenvalues of A are 2, 1 and -1. (In practice, eigenvalues of large matrices are not computed using the characteristic polynomial. Faster and more numerically stable methods are available, for instance the QR decomposition.) Complex eigenvectors Note that if A is a real matrix, the characteristic polynomial will have real coefficients, but not all its roots will necessarily be real. The complex eigenvalues will all be associated to complex eigenvectors. In general, if v1, ..., vm are eigenvectors to different eigenvalues λ1, ..., λm, then the vectors v1, ..., vm are necessarily linearly independent. The spectral theorem for symmetric matrices states that, if A is a real symmetric n-by-n matrix, then all its eigenvalues are real, and there exist n linearly independent eigenvectors for A which all have length 1 and are mutually orthogonal. Our example matrix from above is symmetric, and three mutually orthogonal eigenvectors of A are These three vectors form a basis of R3. With respect to this basis, the linear map represented by A takes a particularly simple form: every vector x in R3 can be written uniquely as and then we have Infinite-dimensional spaces The concept of eigenvectors can be extended to linear operators acting on infinite-dimensional Hilbert spaces or Banach spaces. There are operators on Banach spaces which have no eigenvectors at all. For example, take the bilateral shift on the Hilbert space ; it is easy to see that any potential eigenvector can't be square-summable, so none exist. However, any bounded linear operator on a Banach space V does have non-empty spectrum. The spectrum σ(T) of the operator T : V → V is defined as σ(T)={: (λ1 - T) is not invertible}. Then σ(T) is a compact set of complex numbers, and it is non-empty. When T is a compact operator (and in particular when T is an operator between finite-dimensional spaces as above), the spectrum of T is the same as the set of its eigenvalues. The spectrum of an operator is an important property in functional analysis. See also: spectrum, spectral theorem External links
This article is adapted from from Wikipedia All Wikipedia article text is available under the terms of the GNU Free Documentation License Recent Eigenvector related patents From USPTO: 6711523: Method and apparatus for determining a condition indicator for use in evaluating the health of a component 6711503: Hybrid least squares multivariate spectral analysis methods 6704666: Determining properties of a flow tube and of a fluid flowing through a flow tube of a coriolis flowmeter 6701296: Strain-sensing goniometers, systems, and recognition algorithms 6701006: Apparatus and method for point cloud assembly 6700834: System and method for measuring wave directional spectrum and wave height 6700832: Method and apparatus for passive acoustic imaging using a horizontal line array 6699201: Acoustic window identification 6699189: Ultrasound distortion compensation using blind system identification 6698287: Microgyro tuning using focused ion beams 6697779: Combined dual spectral and temporal alignment method for user authentication by voice 6697778: Speaker verification and speaker identification based on a priori knowledge 6697654: Targeted interference subtraction applied to near-infrared measurement of analytes 6697633: Method permitting increased frequency re-use in a communication network, by recovery of transmitted information from multiple cochannel signals 6697505: Person recognizing apparatus which can reduce time and year changing effects and method therefor 6694283: Eigenvalue quadric surface method for determining when two ellipsoids share common volume for use in spatial collision detection and avoidance 6694279: Methods, apparatus, and computer program products for determining structural motion using mode selective filtering 6694155: Downlink beamforming method 6690816: Systems and methods for tubular object processing 6690814: Image processing apparatus and method 6687672: Methods and apparatus for blind channel estimation based upon speech correlation structure 6687620: Augmented classical least squares multivariate spectral analysis 6687492: System and method for antenna diversity using joint maximal ratio combining 6687336: Line qualification with neural networks 6687188: Underwater telemetry apparatus and method 6683978: Fixed-rate block-based image compression with inferred pixel values 6683689: Method for rapid determination of composition of polycarbonate resin 6683455: Methods for spectral analysis and their applications: spectral replacement 6681032: Real-time facial recognition and verification system 6678690: Retrieving and ranking of documents from database description 6678681: Information extraction from a database 6678450: Optical method for quantum computing 6678389: Method and apparatus for embedding digital information in digital multimedia data 6678048: Information-efficient spectral imaging sensor with TDI 6675145: Method and system for integrated audiovisual speech coding at low bitrate 6675137: Method of data compression using principal components analysis 6675106: Method of multivariate spectral analysis 6674795: System, device and method for time-domain equalizer training using an auto-regressive moving average model 6674526: Methods and apparatus for improving the long-term stability of spectroscopic quantitative analyses 6253181: Speech recognition and teaching apparatus able to rapidly adapt to difficult speech of children and foreign speakers 6252974: Method and apparatus for depth modelling and providing depth information of moving objects 6252540: Apparatus and method for two stage hybrid space-time adaptive processing in radar and communication systems 6246779: Gaze position detection apparatus and method 6246412: Interactive construction and refinement of 3D models from multiple panoramic images 6243599: Methods, systems and computer program products for photogrammetric sensor position estimation 6243492: Image feature extractor, an image feature analyzer and an image matching system 6243415: Process of multisensor equalization allowing multisensor reception in the presence of interference and multiple propagation paths and receiver for the implementation thereof 6240098: Method and device for space division multiplexing of radio signals transmitted in cellular radio communications 6238937: Determining endpoint in etching processes using principal components analysis of optical emission spectra with thresholding |