In numerical optimization , the nonlinear conjugategradientmethod generalizes the conjugategradientmethod to nonlinear optimization . For a quadratic function math displaystyle f x math math displaystyle f x Ax b 2 math The minimum of math f math is obtained when the gradient is 0 math nabla x f 2 A top Ax b 0 math . Whereas linear conjugategradient seeks a solution to the linear equation math displaystyle A top Ax A top b math , the nonlinear conjugategradientmethod is generally used to find the local minimum of a nonlinear function using its gradient math nabla x f math alone. It works when the function is approximately quadratic near the minimum, which is the case when the function ... as in the linear conjugategradientmethod but have been obtained with line searches. The conjugategradientmethod can follow narrow ill conditioned valleys where the steepest descent method slows down ... than a conjugategradient iteration as Newton like methods require computing the Hessian matrix of second derivatives in addition to the gradient. Quasi Newton methods also require more memory to operate see also the limited memory L BFGS method . External links http www.cs.cmu.edu quake papers painless conjugate gradient.pdf An Introduction to the ConjugateGradientMethod Without the Agonizing Pain ... of Scientific Computing , chapter 10, section 6 ConjugateGradient Methods in Multidimensions William ... to minimize, its gradient math nabla x f math indicates the direction of maximum increase. One ... of moving along a subsequent conjugate direction math displaystyle Lambda x n math , where math ... x n math , Compute math displaystyle beta n math according to one of the formulas below, Update the conjugate ... stops. However resetting every iteration turns the method into steepest descent . The algorithm ... automatically. Newton based methods Newton Raphson Algorithm , Quasi Newton methods e.g., BFGS method ... , Cambridge University Press 2nd edition 1992 . Category Optimization methods Category Gradient methods ... more details
definite matrix positive definite . The conjugategradientmethod is an iterative method , so ... equation s. The conjugategradientmethod can also be used to solve unconstrained optimization ... provides a generalization to non symmetric matrices. Various nonlinear conjugategradientmethod s seek ... sub sub . The conjugategradientmethod as a direct method We say that two non zero vectors u and v ... sub k sub . The conjugategradientmethod as an iterative method If we choose the conjugate ... to the solution x sub sub . So, we want to regard the conjugategradientmethod as an iterative method ... used in the Fletcher Reeves nonlinear conjugategradientmethod . Example code in GNU Octave source ... end end source Numerical example To illustrate the conjugategradientmethod, we will complete a simple ... two steps of the conjugategradientmethod beginning with the initial guess math mathbf x 0 begin ... gradientmethod The conjugategradientmethod can theoretically be viewed as a direct method ... the size of the matrix, in the absence of round off error . However, the conjugategradientmethod ..., and the exact solution is never obtained. Fortunately, the conjugategradientmethod can be used ... below. The preconditioned conjugategradientmethod See also Preconditioner In most cases, preconditioning is necessary to ensure fast convergence of the conjugategradientmethod. The preconditioned conjugategradientmethod takes the following form math mathbf r 0 mathbf b mathbf A x 0 math math ... formulation is equivalent to applying the conjugategradientmethod without preconditioning to the system ..., the behavior of the preconditioned conjugategradientmethod may become unpredictable. The flexible preconditioned conjugategradientmethod In numerically challenging applications, sophisticated ... of the convergence of the algorithm presented above. Using the nonlinear conjugategradientmethod ... z k mathrm T mathbf r k math instead of the nonlinear conjugategradientmethod Fletcher Reeves ... more details
In numerical linear algebra , the conjugategradientmethod is an iterative method for numerically solving the System of linear equations linear system math boldsymbol Ax boldsymbol b math where math boldsymbol A math is Symmetric matrix symmetric Positive definite matrix positive definite . The conjugategradientmethod can be derived from several different perspectives, including specialization of the conjugate direction method for Optimization mathematics optimization , and variation of the Arnoldi ... is to document the important steps in these derivations. Derivation from the conjugate direction method Expand section date April 2010 The conjugategradientmethod can be seen as a special case of the conjugate direction method applied to minimization of the quadratic function math f boldsymbol ... above straightforwardly lead to the direct Lanczos method, which turns out to be slightly more complex. The conjugategradientmethod from imposing orthogonality and conjugacy If we allow math ... . math The conjugate direction method In the conjugate direction method for minimizing math f boldsymbol ..., boldsymbol p 2, ldots math are a series of mutually conjugate directions, i.e., math boldsymbol p i mathrm T boldsymbol Ap j 0 math for any math i neq j math . The conjugate direction method is imprecise ... p 1, boldsymbol p 2, ldots math . Specific choices lead to various methods including the conjugategradientmethod and Gaussian elimination . Derivation from the Arnoldi Lanczos iteration see Arnoldi iteration Lanczos iteration The conjugategradientmethod can also be seen as a variant of the Arnoldi Lanczos iteration applied to solving linear systems. The general Arnoldi method In the Arnoldi ... V i boldsymbol y i math . The direct Lanzcos method For the rest of discussion, we assume that math .... R. last2 Stiefel first2 E. title Methods of conjugate gradients for solving linear systems journal ... linear algebra Category Optimization methods Category Gradient methods Category Articles containing ... more details
Multiple issues confusing January 2010 tone January 2010 unreferenced January 2010 wikify January 2011 The conjugate method is a multi faceted method of rotating and linking special exercises that are close in nature to one another. Description of the method The most common template for this method revolves around three methods of weight training used in conjunction with one another. These three methods are Overcoming maximal resistance that causes maximal or near maximal muscle tension maximal effort method . Using considerably less than maximal resistance until fatigue causes one to fail repeated effort method . Using sub maximal weights accompanied by maximal speed dynamic method . External links http www.westside barbell.com Category Weightlifting Category Powerlifting ... more details
In optimization mathematics optimization , gradientmethod is an algorithm to solve problems of the form math min x in mathbb R n f x math with the search directions defined by the gradient of the function at the current point. Examples of gradientmethod are the gradient descent and the conjugategradient . See also col begin col break Gradient descent methodConjugategradientmethod Derivation of the conjugategradientmethod Nonlinear conjugategradientmethod Biconjugate gradientmethod Biconjugate gradient stabilized method References cite book year 1997 title Optimization Algorithms and Consistent Approximations publisher Springer Verlag isbn 0 387 94971 2 author Elijah Polak Optimization algorithms DEFAULTSORT GradientMethod Category First order methods Category Optimization methods Category Numerical linear algebra Category Gradient methods fr Algorithme du gradient ja ... more details
The conjugate residual method is an iterative numeric method used for solving systems of linear equations . It s a Krylov subspace method very similar to the much more popular conjugategradientmethod , with similar construction and convergence properties. This method is used to solve linear equations of the form math mathbf A mathbf x mathbf b math where A is an invertible and Hermitian matrix , and b is nonzero. The conjugate residual method differs from the closely related conjugategradientmethod primarily in that it involves somewhat more computation but is applicable to problems that aren t positive definite, in fact the only requirement is that A be Hermitian matrix Hermitian or, with real numbers, symmetric . This makes the conjugate residual method applicable to problems which intuitively require finding saddle points instead of minima, such as numeric optimization with Lagrange multiplier constraints. Given an arbitrary initial estimate of the solution math mathbf x 0 math , the method is outlined below math mathbf x 0 text Some initial guess , math math mathbf r 0 mathbf b mathbf A x 0 , math math mathbf p 0 mathbf r 0 , math math text Iterate, with k text starting at 0 math math alpha k frac mathbf r k mathrm T mathbf A r k mathbf A p k mathrm T mathbf A p k , math math mathbf x k 1 mathbf x k alpha k mathbf p k , math math mathbf r k 1 mathbf r k alpha k mathbf A p k , math math beta k frac mathbf r k 1 mathrm T mathbf A r k 1 mathbf r k mathrm T mathbf A r k , math math mathbf p k 1 mathbf r k 1 beta k mathbf p k , math math mathbf A p k 1 mathbf A r k 1 beta k mathbf A p k , math math k k 1 , math the iteration may be stopped once math mathbf x k math has been deemed converged. Note that the only difference between this and the conjugategradientmethod is the calculation of math alpha k math and math beta k math plus the optional recursive calculation of math mathbf A p k math at the end . References Yousef Saad , Iterative methods for sparse linear ... more details
math , and the conjugategradientmethod produces the same sequence math x k x k math at half the computational ... to the systems math A x b, , math math x , A b . , math Discussion The biconjugate gradientmethod ... gradient stabilized method , but very important from theoretical point of view. Define the iteration ... math r k r j left 1 P k right math where math i,j k math . The biconjugate gradientmethod now makes ... i mathrm deg left P i right k math , then math v i P i left AM 1 right r k 0 math . see also Conjugategradientmethod Biconjugate gradient stabilized method References cite journal first R. last Fletcher year 1976 title Conjugategradient methods for indefinite systems journal Numerical Analysis ..., and the algorithm takes the form stated above. Properties If math A A , math is Conjugate transpose ... BFb0080109 Numerical linear algebra Category Numerical linear algebra Category Gradient methods de BiCG Verfahren fr M thode du gradient biconjugu ... more details
also Biconjugate gradientmethodConjugategradient squared methodConjugategradientmethod References ...In numerical linear algebra , the biconjugate gradient stabilized method , often abbreviated as BiCGSTAB , is an iterative method developed by Henk van der Vorst H. A. van der Vorst for the numerical solution of nonsymmetric System of linear equations linear system s. It is a variant of the biconjugate gradientmethod BiCG and has faster and smoother convergence than the original BiCG as well as other variants such as the conjugategradient squared method CGS . It is a Krylov subspace method. Algorithmic steps Unpreconditioned BiCGSTAB To solve a linear system math var Ax var var b var , BiCGSTAB starts with an initial guess math var x var sub 0 sub and proceeds as follows math var r var sub 0 sub var b var var Ax var sub 0 sub Choose an arbitrary vector math var r var sub 0 sub such that math var r var sub 0 sub , var r var sub 0 sub 0 , e.g., math var r var sub 0 sub var r var sub 0 sub math var var sub 0 sub var var var var sub 0 sub 1 math var v var sub 0 sub var p var sub 0 sub 0 For math var i var 1, 2, 3, math var sub i sub var var r var sub 0 sub , var r var sub var i var 1 sub math var var var sub i sub var var var sub var i var 1 sub var var var var sub var i var 1 sub math var p sub i sub var var r var sub var i var 1 sub var var var p var sub var i var 1 sub var var sub var i var 1 sub var v var sub var i var 1 sub math var v sub i sub var var Ap var sub var i var sub math var var var sub i sub var var r var sub 0 sub , var v sub i sub var math var s var var r var sub var i var 1 sub var v sub i sub var math var t var var As var math var sub i sub var var t var , var s var var t var , var t var math var x sub i sub var var x var sub var i var 1 sub var p sub i sub var var sub i sub s var If math var x sub i sub var is accurate ... Numerical linear algebra Category Numerical linear algebra Category Gradient methods Category Articles ... more details
Slope other uses In vector calculus , the gradient of a scalar field is a vector field which points ... magnitude is the greatest rate of change. A generalization of the gradient for functions on a Euclidean ..., the scalar field is in black and white, black representing higher values, and its corresponding gradient ...,z math we will assume that the temperature does not change in time . At each point in the room, the gradient ... of the gradient will determine how fast the temperature rises in that direction. Consider a surface whose height above sea level at a point math x, y math is math H x, y math . The gradient ... grade at that point. The steepness of the slope at that point is given by the magnitude of the gradient vector. The gradient can also be used to measure how a scalar field changes in other directions ... will also be 40 . If, instead, the road goes around the hill at an angle the gradient vector , then it will have ... height function math H math is differentiable function differentiable , then the gradient of math ... of the vector. More precisely, when math H math is differentiable, the dot product of the gradient ... H math in the direction of that unit vector. Definition Image Gradient99.png thumb 350px The gradient ... as a projected vector field on the bottom plane The gradient or gradient vector field of a scalar ... math operatorname grad f math is also used for the gradient. The gradient of f is defined to be the vector ... f left frac partial f partial x 1 , dots, frac partial f partial x n right . math Here the gradient ... on a parameter such as time, the gradient often refers simply to the vector of its spatial derivatives only. The gradient of a vector math mathbf f f 1 , f 2 , f 3 math is math nabla mathbf f frac ... partial f 1 , f 2 , f 3 partial x 1 , x 2 , x 3 math . More generally, the gradient may be defined ... dimensional rectangular coordinates The form of the gradient depends on the coordinate system used ... z hat mathbf k math Example For example, the gradient of the function in Cartesian coordinates math ... more details
Image Conjugate Diameters.svg thumb 300px right Two conjugate diameters of an ellipse . Each edge of the bounding parallelogram is Parallel geometry parallel to one of the diameters. In geometry , two diameter s of a conic section are said to be conjugate if each chord geometry chord parallel geometry parallel to one diameter is bisection bisected by the other. For example, two diameters of a circle are conjugate if and only if they are perpendicular . For an ellipse , two diameters are conjugate if and only if the tangent line to the ellipse at the endpoint of one diameter is parallel to the other. Each pair of conjugate diameters of an ellipse has a corresponding tangent parallelogram , sometimes called a bounding parallelogram . In his manuscript De motu corporum in gyrum , and in the Philosophi Naturalis Principia Mathematica Principia , Isaac Newton cites as a lemma mathematics lemma ... . It is possible to Compass and straightedge constructions reconstruct an ellipse from any pair of conjugate ... of his Collection , Pappus of Alexandria gives a method for constructing the axes of an ellipse from a given pair of conjugate diameters. File Drini conjugatehyperbolas.svg thumb right Blue and green hyperbolas are conjugate. A diameter from x,y to &minus x ,&minus y is conjugate to the one from y,x to &minus y ,&minus x . Two hyperbola s are conjugate if they are images of each other in a reflection mathematics reflection across an asymptote . A diameter of one hyperbola is conjugate to its reflection ... to each other. Conjugate diameters of hyperbolas are useful for stating the principle of relativity ... a constant proper time from it. The principle of relativity can be formulated Any pair of conjugate diameters of conjugate hyperbolas can be taken for the axes of space and time . This interpretation ... title Conjugate Diameters of Ellipse http www.cut the knot.org Curriculum Geometry ConjugateDiameters.shtml Conjugate Diameters in Ellipse at cut the knot.org. William Kingdon Clifford 1878 http ... more details
In chemistry , a Lewis conjugate might mean The conjugate base of a Lewis acid or the conjugate acid of a Lewis base A molecule having a conjugated system of bonds in its Lewis structure Disambig ... more details
File Complex conjugate picture.svg right thumb Geometric representation of math z math and its conjugate math bar z math in the complex plane In mathematics , complex conjugates are a pair of complex number s, both having the same real number real part, but with imaginary number imaginary parts of equal magnitude and opposite sign mathematics sign s. ref MathWorld ComplexConjugate Complex Conjugates ref ref MathWorld ImaginaryNumber Imaginary Numbers ref For example, 3 4i and 3 &minus 4i are complex conjugates. The conjugate of the complex number math z math math z a ib, , math where math a math ... 3 2i math math overline 7 7 math math overline i i. math An alternative notation for the complex conjugate is math z math . However, the math bar z math notation avoids confusion with the notation for the conjugate ... coordinate system Complex numbers polar form , the conjugate of math r e i phi math is math r e i ... natural settings, if a complex number provides a solution to a problem, so does its conjugate, such as is the case ... i.e., the conjugate of the conjugate of a complex number z is again that number math z 1 frac overline z left z right 2 math if z is non zero The latter formula is the method of choice to compute the inverse .... Thus, non real roots of real polynomials occur in complex conjugate pairs see Complex conjugate ... z rho e i theta math is given, its conjugate is sufficient to reproduce the parts of the z variable ... z 0 u math determines the line through math z 0 , math in the direction of u. These uses of the conjugate ... complex number s are also explicated by use of complex conjugation. Taking the conjugate transpose ... for quaternion s and coquaternion s the conjugate of math a bi cj dk math is math a bi cj dk math ... is the conjugate transpose operation of complex matrices defined above. It should be remarked ... also Complex conjugate vector space Real structure Complex structure Notes Reflist References Budinich ... maps are discussed in section 3.3 . DEFAULTSORT Complex Conjugate Category Complex numbers bs ... more details
Image Isogonal Conjugate.svg 200px right thumb Isogonal coniugate of P . Image Isogonal Conjugate transform.svg 200px right thumb Isogonal coniugate transformation over the points inside the triangle. In geometry , the isogonal conjugate of a point geometry point P with respect to a triangle ABC is constructed by reflection mathematics reflecting the lines PA , PB , and PC about the angle bisectors of A , B , and C . These three reflected lines concurrent lines concur at the isogonal conjugate of P . This definition applies only to points not on a sideline of triangle ABC . The isogonal conjugate of a point P is sometimes denoted by P . The isogonal conjugate of P is P . The isogonal conjugate of the incentre I is itself. The isogonal conjugate of the orthocentre H is the circumcentre O . The isogonal conjugate of the centroid G is by definition the symmedian symmedian point K . In trilinear coordinates , if X x y z is a point not on a sideline of triangle ABC , then its isogonal conjugate is 1 x 1 y 1 z . For this reason, the isogonal conjugate of X is sometimes denoted by X sup &minus 1 sup . The set S of triangle centers under trilinear product, defined by p q r u v w pu qv rw , is a commutative group, and the inverse of each X in S is X sup &minus 1 sup . As isogonal conjugation is a function, it makes sense to speak of the isogonal conjugate of sets of points, such as lines and circles. For example, the isogonal conjugate of a line is a circumconic and inconic circumconic specifically, an ellipse, parabola, or hyperbola according as the line intersects the circumcircle in 0, 1, or 2 points. The isogonal conjugate of the circumcircle is the line at infinity. Several well known cubics e.g., Thompson cubic, Darboux cubic, Neuberg cubic are self isogonal conjugate ... also Isotomic conjugate Brocard point External links http www.uff.br trianglecenters isogonal conjugate en.html Interactive Java Applet illustrating isogonal conjugate and its properties http mathworld.wolfram.com ... more details
In mathematics , two real number s math p, q 1 math are called conjugate indices if math frac 1 p frac 1 q 1. math Formally, we will also define math q infty math as conjugate to math p 1 math and List of Latin phrases V vice versa vice versa . Conjugate indices are used in H lder s inequality . Also, if math p, q 1 math are conjugate indices, the spaces L sup p sup and L sup q sup are dual space dual to each other see Lp space L sup p sup space . See also Beatty s theorem References A B Antonevich, Linear Functional Equations , Birkh user, 1999. ISBN 3 7643 2931 9. planetmath id 2051 title Conjugate index Category Functional analysis zh ... more details
In group theory , the conjugate closure of a subset S of a group mathematics group G is the subgroup of G generating set of a group generated by S sup G sup , i.e. the closure of S sup G sup under the group operation, where S sup G sup is the conjugates of the elements of S S sup G sup g sup &minus 1 sup sg g &isin G and s &isin S The conjugate closure of S is denoted S sup G sup or S sup G sup . The conjugate closure of any subset S of a group G is always a normal subgroup of G in fact, it is the smallest by inclusion normal subgroup of G which contains S . For this reason, the conjugate closure is also called the normal closure of S or the normal subgroup generated by S . The normal closure can also be characterized as the intersection set theory intersection of all normal subgroups of G which contain S . Any normal subgroup is equal to its normal closure. The conjugate closure of a singleton set singleton subset a of a group G is a normal subgroup generated by a and all elements of G which are conjugate to a . Therefore, any simple group is the conjugate closure of any non identity group element. The conjugate closure of the empty set math varnothing math is the trivial group . Contrast the normal closure of S with the normalizer of S , which is for S a group the largest subgroup of G in which S itself is normal. This need not be normal in the larger group G , just as S need not be normal in its conjugate normal closure. References cite book title Handbook of Computational Group Theory author Derek F. Holt coauthors Bettina Eick, Eamonn A. O Brien publisher CRC Press year 2005 isbn 1584883723 pages 73 Category Group theory algebra stub zh ... more details
Unreferenced date December 2009 In differential geometry , conjugate points are, roughly, points that can almost be joined by a 1 parameter family of geodesic s. For example, on a Spherical geometry sphere , the north pole and south pole are connected by any Meridian geography meridian . Definition Suppose p and q are points on a Riemannian manifold , and math gamma math is a geodesic that connects p and q . Then p and q are conjugate points along math gamma math if there exists a non zero Jacobi field along math gamma math that vanishes at p and q . Recall that any Jacobi field can be written as the derivative of a geodesic variation see the article on Jacobi field s . Therefore, if p and q are conjugate along math gamma math , one can construct a family of geodesics which start at p and almost end at q . In particular, if math gamma s t math is the family of geodesics whose derivative in s at math s 0 math generates the Jacobi field J , then the end point of the variation, namely math gamma s 1 math , is the point q only up to first order in s . Therefore, if two points are conjugate, it is not necessary that there exist two distinct geodesics joining them. Examples On the sphere math S 2 math , antipodal points are conjugate. On math mathbb R n math , there are no conjugate points. On Riemannian manifolds with non positive sectional curvature , there are no conjugate points. See also Cut locus Riemannian manifold cut locus Jacobi field DEFAULTSORT Conjugate Points Category Riemannian geometry ... more details
In geometry , the isotomic conjugate of a point P not on a sideline of triangle ABC is constructed as follows Let A nowiki nowiki , B nowiki nowiki , C nowiki nowiki be the points in which the lines AP , BP , CP meet the lines BC , CA , AB , respectively. Reflect A nowiki nowiki B nowiki nowiki C nowiki nowiki in the midpoints of sides BC , CA , AB to obtain points A , B , C , respectively. The lines AA , BB , CC meet at a point this can be proved using Ceva s theorem , and this point is called the isotomic conjugate of P . If Trilinear coordinates trilinears for P are p q r , then trilinears for the isotomic conjugate of P are a sup &minus 2 sup p sup &minus 1 sup b sup &minus 2 sup q sup &minus 1 sup c sup &minus 2 sup r sup &minus 1 sup . The isotomic conjugate of the centroid of triangle ABC is the centroid itself. Isotomic conjugates of lines are circumconics, and conversely, isotomic conjugates of circumconics are lines. This property holds for isogonal conjugates as well. See also Isogonal conjugate References Robert Lachlan, An Elementary Treatise on Modern Pure Geometry , Macmillan and Co., 1893, page 57. Category Triangles de Isotomisch konjugierte Punkte nl Isotomische verwantschap zh ... more details
About conjugates in field extensions Conjugate disambiguation Conjugation In algebra , a conjugate of an element in a quadratic extension field of a field K is its image under the unique non identity automorphism of the extended field that fixes K . If the extension is generated by a square root of an element r of K , then the conjugate of math a b sqrt r math is math a b sqrt r math for math a,b in K math . In the case of the field C of complex number s, the complex conjugate of a bi is a bi . Forming the sum or product of any element of the extension field with its conjugate always gives an element of K . This can be used to rewrite a quotient of numbers in the extended field so that the denominator lies in K , by multiplying numerator and denominator by the conjugate of the denominator. This process is called rationalization of the denominator, in particular if K is the field Q of rational numbers. Uses Differences of squares An expression of the form math a 2 b 2 , math can be factored to give math a b a b , math where one factor is the conjugate of the other. This can be useful when trying to Rationalisation mathematics rationalize a denominator containing a radical. Rationalizing radicals in the denominator An irrational denominator of the form math a b sqrt r math can be made rational by multiplying numerator and denominator by the conjugate math a b sqrt r math , so that the denominator becomes math a 2 ab sqrt r ab sqrt r b 2r math , or math a 2 b 2r math . math frac 1 a b sqrt r frac 1 a b sqrt r cdot frac a b sqrt r a b sqrt r frac a b sqrt r a 2 b 2r math Here is an example math frac 1 2 2 sqrt 3 frac 1 2 2 sqrt 3 cdot frac 2 2 sqrt 3 2 2 sqrt 3 frac 2 2 sqrt 3 2 2 2 2 cdot 3 frac 2 sqrt 3 2 8 frac sqrt 3 1 4 , math See also Complex conjugate External links http www.mathwords.com r rationalizing the denominator.htm Rationalizing the Denominator from Mathwords.com http www.blc.edu fac rbuelow common glossarya m.htm conjugate Math glossary from Bethany Lutheran ... more details
th entry, for 1 &le i &le n and 1 &le j &le m , and the overbar denotes a scalar complex conjugate . The complex conjugate of math a bi math , where a and b are reals, is math a bi math . This definition ... the matrix with complex conjugated entries. Other names for the conjugate transpose of a matrix are Hermitian conjugate , or transjugate . The conjugate transpose of a matrix A can be denoted by any ... entries, and thus the conjugate transpose is denoted by math mathbf A T , math or math mathbf ... The conjugate transpose can be motivated by noting that complex numbers can be usefully represented ... by 2n matrix of real numbers. The conjugate transpose therefore arises very naturally as the result .... Properties of the conjugate transpose A B sup sup A sup sup B sup sup for any two matrices A and B ... A . Here r sup sup refers to the complex conjugate of r . AB sup sup B sup sup A sup sup for any m by n ... spaces can thus be seen as a generalization of the conjugate transpose of matrices. Another generalization ... conjugate linear map as well as the transpose of a linear map transposed linear map are defined, and we may thus take the conjugate transpose of A to be the complex conjugate of the transpose of A . It maps the conjugate dual space dual of W to the conjugate dual of V . See also Hermitian adjoint Hermitian conjugate Adjugate matrix External links MathWorld urlname ConjugateTranspose title Conjugate Transpose planetmath reference id 4382 title Conjugate transpose Category Linear algebra ... more details
For conjugate variables in context of thermodynamics Conjugate variables thermodynamics Conjugate variables are pairs of variables mathematically defined in such a way that they become Fourier transform dual mathematics duals of one another, ref http www.aip.org history heisenberg p08a.htm Heisenberg Quantum Mechanics, 1925 1927 The Uncertainty Relations ref ref http www.springerlink.com content r40472577250313r Some remarks on time and energy as conjugate variables ref or more generally are related through Pontryagin duality . The duality relations lead naturally to an uncertainty in physics called the Heisenberg uncertainty principle relation between them. In mathematical terms, conjugate variables are part of a symplectic basis, and the uncertainty principle corresponds to the symplectic form . Examples There are many types of conjugate variables, depending on the type of work a certain system is doing or is being subjected to . Examples of canonically conjugate variables include the following Time and frequency the longer a musical note is sustained, the more precisely we know its frequency but it spans more time . Conversely, a very short musical note becomes just a click, and so one can t know its frequency very accurately. Doppler effect Doppler and range the more we know about how far away a radar target is, the less we can know about the exact velocity of approach or retreat, and vice versa. In this case, the two dimensional function of doppler and range is known as a radar ambiguity function or radar ambiguity diagram . Surface energy dA surface tension A surface area Elastic stretching FdL F elastic force L length stretched Derivatives of action In classical physics, the derivatives of action physics action are conjugate variables to the quantity with respect ... density at that event. See also Canonical coordinates Notes Reflist DEFAULTSORT Conjugate ... pl Zmienne sprz one simple Conjugate variables zh ... more details
Conjugate coding is a cryptographic tool, introduced by Stephen Wiesner ref http portal.acm.org citation.cfm?id 1008908.1008920 ref in the sixties . Because its publication has been surprisingly rejected, it was developed to the world of public key cryptography in the eighties as Oblivious Transfer , first by Michael O. Rabin Rabin and then by Shimon Even Even . It is used in the field of Quantum Computing . References references Category Cryptography Category Information theory crypto stub ... more details
for geometric conjugate points Projective harmonic conjugates In mathematics , a function math u x, ,y math defined on some open domain math Omega subset R 2 math is said to have as a conjugate a function math v x, ,y math if and only if they are respectively real and imaginary part of a holomorphic function math f z math of the complex variable math z x iy in Omega. math That is, math v math is conjugated to math u math if math f z u x,y iv x,y math is holomorphic on math Omega. math As a first consequence of the definition, they are both harmonic function harmonic real valued functions on math Omega math . Moreover, the conjugate of math u, math if it exists, is unique up to an additive constant. Also, math u math is conjugate to math v math if and only if math v math is conjugate to math u math . Equivalently, math v math is conjugate to math u math in math Omega math if and only if math u math and math v math satisfy the Cauchy Riemann equations in math Omega. math As an immediate consequence of the latter equivalent definition, if math u math is any harmonic function on math Omega subset R 2, math the function math u y math is conjugate to math u x math , for then the Cauchy Riemann ... math in math Omega, math in which case a conjugate of math u math is, of course, math scriptstyle mathrm Im ,f x iy . math So any harmonic function always admits a conjugate function whenever its domain is simply connected , and in any case it admits a conjugate locally at any point of its domain. There is an operator ... harmonic conjugate v putting e.g. v x sub 0 sub 0 on a given x sub 0 sub in order to fix the indeterminacy of the conjugate up to constants . This is well known in applications as essentially the Hilbert ... operator s. Conjugate harmonic functions and the transform between them are also one of the simplest ... conjugate of x is y , and the lines of constant x and constant y are orthogonal. Conformality ... occurrence of the term harmonic conjugate in mathematics , and more specifically ... more details
f X to mathbb R cup infty math taking values on the extended real number line the convex conjugate ... supporting hyperplanes . http maze5.net ?page id 733 Examples The convex conjugate of an affine ... star left x right begin cases b, & x a infty, & x ne a. end cases math The convex conjugate of a power ... math where math tfrac 1 p tfrac 1 q 1. math The convex conjugate of the absolute value function ... x right 1. end cases math The convex conjugate of the exponential function math f x , beta cdot ... , & x 0 0 , & x 0 infty , & x 0. end cases math Convex conjugate and Legendre transform of the exponential function agree except that the domain mathematics domain of the convex conjugate is strictly ... the convex conjugate math begin align f star p int 0 p F 1 q , dq & p 1 F 1 p operatorname E left ..., math f text inc f math for &fnof nondecreasing. Properties The convex conjugate of a closed convex function is again a closed convex function. The convex conjugate of a polyhedral convex function a convex ... math . Here math f le g iff forall x, f x le g x . math Biconjugate The convex conjugate of a function is always lower semi continuous . The biconjugate math f math the convex conjugate of the convex conjugate is also the closed convex hull , i.e. the largest lower semi continuous convex function with math ... and lower semi continuous. Fenchel s inequality For any function f and its convex conjugate ... of the function is the maximizing argument to compute the convex conjugate math f prime ... conjugate f sup sup is symmetric with respect to G . Infimal convolution The infimal convolution of two ... more details
family as the prior probability distribution p , the prior and posterior are then called conjugate distributions, and the prior is called a conjugate prior for the likelihood. For example, the Normal distribution Gaussian family is conjugate to itself or self conjugate with respect to a Gaussian ... conjugate prior , were introduced by Howard Raiffa and Robert Schlaifer in their work on Bayesian decision ... of Mathematics , http jeff560.tripod.com c.html conjugate prior distributions . Electronic document ... with different parameter values . Such a choice is a conjugate prior . A conjugate prior is an algebraic ... integration may be necessary. Further, conjugate priors may give intuition, by more transparently ... conjugate priors. See Gelman et al. ref name gelman et al Andrew Gelman , John B. Carlin, Hal S. Stern ... X. ref for a catalog. Example The form of the conjugate prior can generally be determined by inspection ... q math . In fact, the usual conjugate prior is the beta distribution with math p q q alpha 1 1 q beta ... of conjugate priors that the dimensionality of the hyperparameters is one greater than that of the parameters ... distribution , conjugate prior of the covariance matrix of a multivariate normal distribution ... extra piece of information as it comes. Interpretations Analogy with eigenfunctions Conjugate priors ... to the posterior as an operator. In both eigenfunctions and conjugate priors, there is a finite ... , one can easily analyze how a convex combination of conjugate priors evolves under conditioning this is called using a hyperprior , and corresponds to using a mixture density of conjugate priors, rather than a single conjugate prior. Dynamical system One can think of conditioning on conjugate ... on time, but rather on data over time. Table of conjugate distributions Let n denote the number of observations. If the likelihood function belongs to the exponential family , then a conjugate prior exists, often also in the exponential family see Exponential family Bayesian estimation conjugate ... more details
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