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Topic: Conjugate transpose

Related Topics

Hermitian conjugate

Diagonal matrix

Inverse matrix

Normal matrix

Matrix (mathematics)

Hermitian matrix

Unitary matrix

Complex conjugate

Spectral theorem

Lower triangular matrix

Pseudoinverse

Differential operator

Singular value decomposition

List of linear algebra topics

Symmetric matrix

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	Transpose
		Informally, the transpose of a square matrix is obtained by reflecting at the main diagonal (that runs from the top left to bottom right of the matrix).
		This shows that the transpose is a linear map from the space of all m-by-n matrices to the space of all n-by-m matrices.
		A square matrix whose transpose is equal to itself is called a symmetric matrix, i.e.
	www.ebroadcast.com.au /lookup/encyclopedia/tr/Transpose.html (449 words)

	Conjugate transpose: Definition and Links by Encyclopedian.com (via CobWeb/3.1 planetlab2.cs.unc.edu) (Site not responding. Last check: 2007-10-19)
		Generalizations Taking the conjugate transpose (or adjoint) of complex matrices generalizes complex...quaternions : the conjugate of a + bi + cj + dk is a - bi - cj - dk.
		In mathematics, the conjugate transpose or adjoint of an m-by-n matrix A with complex entries is the n-by-m matrix A
		It is useful to think of square complex matrices as "generalized complex numbers", and of the conjugate transpose as a generalization of complex conjugation.
	www.encyclopedian.com.cob-web.org:8888 /ad/Adjoint-matrix.html (409 words)

	PlanetMath: conjugate transpose
		It is clear that for real matrices, the conjugate transpose coincides with the transpose.
		is also used for the conjugate transpose [2].
		This is version 7 of conjugate transpose, born on 2003-06-21, modified 2006-09-13.
	planetmath.org /encyclopedia/ConjugateTranspose.html (109 words)

	PlanetMath: transpose
		The transpose is a linear mapping from the vector space of matrices to itself.
		This is version 8 of transpose, born on 2001-11-20, modified 2005-03-18.
		Object id is 975, canonical name is Transpose.
	planetmath.org /encyclopedia/Transpose.html (115 words)

	Conjugate transpose - Wikipedia, the free encyclopedia
		It is often useful to think of square complex matrices as "generalized complex numbers", and of the conjugate transpose as a generalization of complex conjugation.
		Another generalization is available: suppose A is a linear map from a complex vector space V to another W, then the complex conjugate linear map as well as the 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 of W to the conjugate dual of V.
	en.wikipedia.org /wiki/Conjugate_transpose (503 words)

	Quaternion
		consisting of quaternions with real part equal to zero: it is not hard to see that the conjugation by a unit quaternion (a quaternion of absolute value 1) with real part cos t is a rotation by an angle 2t, the axis of the rotation being the direction of the imaginary part.
		The conjugate of a quaternion corresponds to the conjugate transpose of the matrix.
		In this representation, the conjugate of a quaternion corresponds to the transpose of the matrix.
	www.ebroadcast.com.au /lookup/encyclopedia/qu/Quaternian.html (1193 words)

	Math Forum - Ask Dr. Math
		The transpose of a matrix is gotten by flipping everything across the main diagonal.
		So for example, the transpose of: [ a b c ] T [ a d g ] A = [ d e f ] is A = [ b e h ] [ g h i ] [ c f i ] The conjugate is gotten by replacing every element by its complex conjugate.
		For example, the matrix: [ 0 i ] [ i 0 ] is unitary, since its transpose is itself, and the complex conjugate of the transpose is: [ 0 -i ] [ -i 0 ] and when you multiply that matrix by the one above, you get the identity.
	mathforum.org /library/drmath/view/51485.html (377 words)

	Stata help for m2_op_transpose
		A' returns the transpose of A or, if A is complex, the conjugate transpose.
		The transpose operator cannot fail, but it is easy to use it incorrectly when working with complex quantities.
		If it had been, then rather than telling the story of the man who was bitten by conjugate transpose when he only wanted the transpose, we would have told the story of the woman who was bitten by the transpose when she needed the conjugate transpose.
	www.stata.com /help.cgi?m2_op_transpose (338 words)

	Engineering and Scientific Subroutine Library for AIX Version 3 Release 3: Guide and Reference - Use Considerations
		In every case, the input arrays for the matrix, its transpose, or its conjugate transpose should be stored in the original untransposed form.
		If you want to compute the transpose or the conjugate transpose of a matrix operation--that is, the output stored in matrix C--you should use the matrix identities described in Special Usage for each subroutine description.
		This accomplishes the transpose or conjugate transpose as part of the multiply operation.
	www.ncsa.uiuc.edu /UserInfo/Resources/Hardware/IBMp690/IBM/usr/lpp/essl.html.en_US/html/essl118.html (236 words)

	def. of unitary
		if the matrices are real then adjoint is just transpose - part of the definition for unitary is that it is complex, and preserves an hermitian form (which is often how you'll see it defined).
		I don't see how this makes "adjoint" and "transpose conjugate" different (since the conjugate of a real number is itself).
		I was just taught that "adjoint" always means "transpose" and "complex conjugate" and I would like to be aware of any variation in usage for the sake of future conversations with the math people.
	www.physicsforums.com /showthread.php?t=19115 (1091 words)

	Transpose - Wikipedia, the free encyclopedia
		The transpose is a linear map from the space of m × n matrices to the space of all n × m matrices.
		The transpose of a scalar is the same scalar.
		is the (j, i) element of AB by the definition of transpose.
	en.wikipedia.org /wiki/Transpose (508 words)

	Rechenzentrum - Beschreibung der einzelnen Routinen
		is a symmetric matrix, and trans() represents the transpose of the vectors.
		Here trans(A) is the transpose of the matrix, and ctrans(A) is the conjugate transpose of the matrix.
		where ctrans(y) is the conjugate transpose of the vector.
	www.rz.uni-karlsruhe.de /produkte/1700.php (2804 words)

	32.7 Other Notions of Diagonalizability
		However it does not change if you interchange row and column and also take the complex conjugate, since the asymmetry lies in taking the complex conjugate of the row and not the column.
		The answer now is that any matrix that is its own transpose complex conjugate will have this property: which implies if M is n by n, M has n real eigenvalues and an orthonormal basis of eigenvectors.
		This condition is again preserved under unitary transformations, and it is a property of diagonal matrices, since all diagonal matrices commute with one another, so it is definitely necessary.
	www-math.mit.edu /18.013A/HTML/chapter32/section07.html (757 words)

	[No title]
		The inverse of a 1 x 1 matrix (other than 0) is the 1 x 1 matrix with 1/ that entry, which you could find, if you really wanted to, by row reducing.
		The transpose of a 1 x 1 matrix just moves the entry a_11 back to a_11; every number is a symmetric matrix.
		When working with matrices whose entries are complex, the right thing to use instead of the transpose of the matrix is the "conjugate transpose" of the matrix, which is just the transpose but also taking the complex conjugate of every entry--replace a + bi with a - bi.
	math.berkeley.edu /~hthall/54QA.txt (3255 words)

	Matrices
		We may say that the ``rows and columns are interchanged'' by the transpose operation, and transposition can be visualized as ``flipping'' the matrix about its main diagonal.
		The transpose of a complex matrix is normally defined to include conjugation.
		The conjugating transpose operation is called the Hermitian transpose.
	ccrma.stanford.edu /~jos/mdft/Matrices.html (184 words)

	Vector and Matrix Arithmetic
		Conjugate transpose is sometimes also called Hermitian conjugate.
		Matrices that are invariant with respect to Hermitian conjugation are called Hermitian matrices.
		A normal, i.e., not conjugate, transpose is invoked by
	beige.ucs.indiana.edu /P573/node104.html (1137 words)

	Complex conjugate (Site not responding. Last check: 2007-10-19)
		In mathematics, the complex conjugate of a complex number is given by changing the sign of the imaginary part.
		Taking the conjugate transpose (or adjoint) of Complex matrices generalizes Complex conjugation.
		One example of this notion is the conjugate transpose operation of Complex matrices defined above.
	complex-conjugate.iqnaut.net (403 words)

	Complete Inner Product Space (Site not responding. Last check: 2007-10-19)
		The inner-product of two quaternions is defined here as the transpose (or conjugate) of the first quaternion multiplied by the second.
		The conjugate of the square of the norm equals the square of the norm of the two terms reversed.
		In a similar fashion, it is hoped that because the product of a transpose of a quaternion with a quaternion has the properties of a complete inner product space, the power of the mathematical field of quaternions can be used to solve a wide range of problems in quantum mechanics.
	www.theworld.com /~sweetser/quaternions/quantum/bracket/bracket.html (774 words)

	Linear Algebra Glossary (Site not responding. Last check: 2007-10-19)
		The transpose of a circulant matrix is a circulant matrix.
		On the other hand, if N is very large, and a suitable preconditioner is used, it is possible to treat the conjugate gradient method as "essentially" an iterative method, in the sense that it may be the case that a good approximate solution can be found by stopping the algorithm early.
		The conjugate matrix of a complex matrix A, denoted by A* or conjugate (A), is the matrix obtained by replacing each entry of A by its complex conjugate.
	www.csit.fsu.edu /~burkardt/papers/linear_glossary.html (13553 words)

	Conjugate transpose - Wikipedia, the free encyclopedia (via CobWeb/3.1 planetlab2.cs.unc.edu) (Site not responding. Last check: 2007-10-19)
		In mathematics, the conjugate transpose or adjoint matrix of an m-by-n matrix A with complex entries is the n-by-m matrix A
		The conjugate transpose of a matrix A can be denoted by any of these symbols:
		The conjugate transpose can be motivated by noting that complex numbers can be usefully represented by 2×2 skew-symmetric matrices, obeying matrix addition and multiplication:
	en.wikipedia.org.cob-web.org:8888 /wiki/Conjugate_transpose (513 words)

	NMath Matrix User's Guide - 5.1 QR Decompositions
		PTx(), QTx(), and RTx() compute the inner product of the transpose of a component matrix and a given vector, or conjugate transpose for complex types.
		QTM() uses the transpose of Q, or conjugate transpose for complex types.
		RTInvx() uses the transpose of R, or conjugate transpose for complex types.
	www.centerspace.net /doc/NMath/Matrix/user/decompositions2.html (607 words)

	The On-Line Encyclopedia of Integer Sequences (Site not responding. Last check: 2007-10-19)
		Conjectured number of ordered trees on n edges for which the conjugate and transpose commute.
		The conjugate of an ordered tree is given by flipping it over, while its transpose is given by flipping over the corresponding binary tree.
		A list of ordered trees for which the conjugate and transpose commute, counted by this sequence, is given in Exercise 17, Sec.
	www.research.att.com /~njas/sequences/A123050 (273 words)

	First conjugate-gradient program
		The conjugate-gradient program can be divided into two parts: an inner part that is used almost without change over a wide variety of applications, and an outer part containing the initializations.
		# setup of conjugate gradient descent, minimize SUM rr(i)*2 # nx # rr(i) = yy(i) - sum aaa(i,j) x(j) # j=1 subroutine cgmeth(nx,x, nr,yy,rr, aaa, niter) integer i, iter, nx, nr, niter real x(nx), yy(nr), rr(nr), aaa(nr,nx) temporary real dx(nx), sx(nx), dr(nr), sr(nr) do i= 1, nx x(i) = 0.
		Then it loops over iterations, invoking matrix multiply, conjugate transpose multiply, and the conjugate-gradient stepper.
	sepwww.stanford.edu /sep/prof/pvi/ls/paper_html/node16.html (344 words)

	Maths - Group Theory - Martin Baker
		= the conjugate transpose of U (the conjugate transpose is also known as adjoint matrix, adjugate matrix, hermitian adjoint or hermitian transpose).
		The conjugate matrix is the complex conjugate of each element.
		The group SU(2) is equivalent to (is isomorphic to) the set of unit length quaternions with multiplication.
	www.euclideanspace.com /maths/algebra/groups/index.htm (1632 words)

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