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Hermitian (including recent related patents.)
HermitianA number of mathematical entities are named Hermitian, after the mathematician Charles Hermite. A Hermitian matrix is a square matrix with complex entries so that the matrix is equal to its own conjugate transpose - that is, if the element in the ith row and jth column is equal to the complex conjugate of the element in the jth row and ith column, for all indices i and j: The conjugate transpose of a matrix is also called its adjoint, and a synonym for Hermitian is self-adjoint. Here is an example of a Hermitian matrix: 2-i&1\end{bmatrix}"> If all the entries of a matrix are real, then it is Hermitian if and only if it is a symmetric matrix. Every Hermitian matrix is normal, and the finite-dimensional spectral theorem applies. It says that any Hermitian matrix can be diagonalized by a unitary matrix, and that the resulting diagonal matrix has only real entries. This means that all eigenvalues of a Hermitian matrix are real, and, moreover, eigenvectors with distinct eigenvalues are orthogonal. It is possible to find an orthonormal basis of Cn consisting only of eigenvectors. If the eigenvalues of a Hermitian matrix are all positive, then the matrix is positive definite. Hermitian operators A continuous linear operator A: H → H on a Hilbert space A is called Hermitian or self-adjoint if (x,Ay) = (Ax,y) for all elements x and y of H. Here, the parentheses denote the inner product given on H. This definition agrees with the one given above if we take as H the Hilbert space Cn with the standard dot product and interpret a square matrix as a linear operator on this Hilbert space. It is however much more general as there are important infinite-dimensional Hilbert spaces. The spectrum of any Hermitian operator is real; in particular all its eigenvalues are real. For any two Hermitian operators A: H → H and B: H → H, and any element x of H holds the Cauchy-Bunyakovski-Schwarz inequality ( Ax, Ax ) ( Bx, Bx ) ≥ ( ABx, x ) ( x, ABx ) = ( BAx, x ) ( x, BAx ), und consequently the Robertson-Schrödinger relation: ( Ax, Ax ) ( Bx, Bx ) ≥ 1/4 ( (AB - BA)x, x ) ( x, (AB - BA)x ). A version of the spectral theorem also applies to Hermitian operators; while the eigenvectors to different eigenvalues are orthogonal, in general it is not true that the Hilbert space H admits an orthonormal basis consisting only of eigenvectors of the operator. In fact, Hermitian operators need not have any eigenvalues or eigenvectors at all. In the mathematical formulation of quantum mechanics, one considers even more general Hermitian operators: they are only defined on a dense subspace of a Hilbert space and don't have to be continuous. For example, consider the complex Hilbert space L2[0,1] and the differential operator A = d2 / dx2, defined on the subspace consisting of all differentiable functions f : [0,1] → C with f(0) = f(1) = 0. Then integration by parts easily proves that A is Hermitian. Its eigenfunctions are the sinusoids sin(nπx) for n = 1,2,..., with the real eigenvalues n2π2; the well-known orthogonality of the sine functions follows as a consequence of the Hermitian property. Another example: the complex Hilbert space L2(R), and the operator which multiplies a given function by x: Af(x) = xf(x) It is defined on the space of all L2 functions for which the right-hand-side is square-integrable. A is a Hermitian operator without any eigenvalues and eigenfunctions.This article is adapted from from Wikipedia All Wikipedia article text is available under the terms of the GNU Free Documentation License Recent Hermitian related patents From USPTO: 6717995: Method for correcting DC offsets in a receiver 6717979: Method for estimating a direction of arrival 6717406: Parallel magnetic resonance imaging techniques using radiofrequency coil arrays 6714609: Co-channel interference in a receiver 6714527: Multiuser detector for variable spreading factors 6711528: Blind source separation utilizing a spatial fourth order cumulant matrix pencil 6707864: Simplified block linear equalizer with block space time transmit diversity 6704664: Fatigue sensitivity determination procedure 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 6699189: Ultrasound distortion compensation using blind system identification 6697633: Method permitting increased frequency re-use in a communication network, by recovery of transmitted information from multiple cochannel signals 6694165: Method for ultra-fast MR fluoroscopy 6694155: Downlink beamforming method 6690747: Method for reference signal generation in the presence of frequency offsets in a communications station with spatial processing 6690739: Method for intersymbol interference compensation 6690717: Multi-tone transceiver system using two steps of DMT-CMFB 6690660: Adaptive algorithm for a Cholesky approximation 6687292: Timing phase acquisition method and device for telecommunications systems 6687291: Time-domain equalizer of cascaded filters for VDSL 6675187: Pipelined linear array of processor elements for performing matrix computations 6671313: Time-weighted transmission channel estimation 6668161: Determining a spatial signature using a robust calibration signal 6667830: Super-resolution microscope system and method for illumination 6665771: Intra-disk swapping of data storage volumes 6665349: Filtered multitone transmission application to DSL technologies 6664914: Ground penetrating radar 6664913: Lossless coding method for waveform data 6658234: Method for extending the effective dynamic range of a radio receiver system 6658047: Adaptive channel equalizer 6654719: Method and system for blind separation of independent source signals 6654590: Determining a calibration function using at least one remote terminal 6650716: Space-time receiver structure for digital communication systems 6650683: Surface emitting semiconductor laser 6647367: Noise suppression circuit 6646593: Process for mapping multiple-bounce ghosting artifacts from radar imaging data 6640145: Media recording device with packet data interface 6633617: Device and method for compensating or creating doppler effect using digital signal processing 6625459: Method and apparatus for efficient determination of channel estimate and baud frequency offset estimate 6622118: System and method for comparing signals 6618481: Method for improving acoustic sidetone suppression in hands-free telephones 6615024: Method and apparatus for determining signatures for calibrating a communication station having an antenna array 6614616: Determining seek times 6603827: Method and apparatus for digital symbol detection using medium response estimates 6603809: Apparatus, and associated method, for forming a signal for communication upon a fading channel 6600446: Cascadable architecture for digital beamformer 6598009: Method and device for obtaining attitude under interference by a GSP receiver equipped with an array antenna 6597745: Reduced complexity multicarrier precoder 6597678: Radio communication system using adaptive array antenna |