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	Unitary matrix - Wikipedia, the free encyclopedia
		A unitary matrix in which all entries are real is the same thing as an orthogonal matrix.
		It follows from the isometry property that all eigenvalues of a unitary matrix are complex numbers of absolute value 1 (i.e.
		All unitary matrices are normal, and the spectral theorem therefore applies to them.
	en.wikipedia.org /wiki/Unitary_matrix (208 words)

	Matrix Manual: Special Matrices
		A symmetric or Hermitian Hessenberg matrix is tridiagonal.
		A is symmetric iff it is congruent to a diagonal matrix.
		Any square matrix may be uniquely decomposed as the sum of a symmetric matrix and a skew-symmetric matrix.
	www.psi.utoronto.ca /matrix/special.html (2192 words)

	GLOSSARY (Site not responding. Last check: 2007-11-06)
		A band matrix is one whose entries are all zero except for the diagonal and a few of the immediately adjacent diagonals, whose entries may be nonzero.
		An elementary matrix is one which, when pre-multiplying another matrix, interchanges two rows, or multiplies one row by a nonzero constant, or adds a multiple of one row to another.
		The transpose of a matrix is the matrix obtained by switching the pairs of entries A(I,J) and A(J,I).
	www.psc.edu /general/software/packages/eispack/glossary.html (988 words)

	Matrix Reference Manual: Special Matrices
		A 3#3 orthogonal matrix is either a rotation matrix or else a rotation matrix plus a reflection in the plane of the rotation according to whether it is proper or improper.
		A Rotation matrix is orthogonal with a determinant of +1.
		A signature matrix is a diagonal matrix whose diagonal entries are all +1 or -1.
	www.ee.ic.ac.uk /hp/staff/dmb/matrix/special.html (4190 words)

	Linear Algebra Glossary
		The inverse of a (nonsingular) circulant matrix is a circulant matrix.
		The transpose of a circulant matrix is a circulant matrix.
		The vector (1,1,...,1) is an eigenvector of the matrix, with eigenvalue equal to the sum of the entries of a row.
	orion.math.iastate.edu /burkardt/papers/linear_glossary.html (13535 words)

	On the Characteristic Polynomial of a Random Unitary Matrix (Site not responding. Last check: 2007-11-06)
		On the Characteristic Polynomial of a Random Unitary Matrix
		The analogy between the characteristic polynomial of a random unitary matrix and Riemann's zeta function was first studied by Keating and Snaith in \cite{KS1}.
		The Keating-Snaith conjecture and the derivatives conjecture are unified as being special cases of one particular calculation: the discrete moments of the modulus of zeta, evaluated close to its non-trivial zeros.
	www.aimath.org /~hughes/PhD_abstract.html (272 words)

	► » Unitary matrix. (Site not responding. Last check: 2007-11-06)
		cos(theta) in such a way that it will be explicitly unitary.
		and matrix will describe rotation in 3 dimensional space.
		and form an antisymmetric unitary group as well.
	www.science-chat.org /Unitary-matrix--6930642.html (172 words)

	Math 528A: Banach and Hilbert Space (Site not responding. Last check: 2007-11-06)
		The unitary equivalence is given by a unitary operator.
		In the matrix case this is a unitary matrix given by an orthonormal basis of eigenvectors.
		However it plays a special role in quantum physics, where the basic equations are dictated by a commutation relation between self-adjoint operators.
	math.arizona.edu /~faris/hilbert.html (206 words)

	Re: What is the "Spin" of a partile (Site not responding. Last check: 2007-11-06)
		Spin 1/2 is described by a pair of complex numbers, and when you make (the same) change of coordinate, it is transformed to the new coordinates by being multiplies by a different matrix (a special unitary matrix, a member of SU(2)).
		As you run the orthogonal matrices along a path that circles their manifold and returns to its start, the corresponding special unitary matrices make two circuits around their manifold (which is the 3-sphere).
		This is the mathematical equivalent of the twice-aroundness of the quantum spin 1/2.
	superstringtheory.com /forum/basicboard/messages3/120.html (379 words)

	Matrix Manual: Special Matrices
		E is an essential matrix iff its singular values are k, k and 0 for some k>0.
		[22]: A 22 orthogonal matrix is either a Givens rotation or a Givens reflection.
		Any matrix A has a unique decomposition A = S + K where S is Hermitian and K is skew-hermitian.
	www.ee.uwa.edu.au /~roberto/teach/matrix/special.html (2558 words)

	The Fourier Matrix
		The Fourier matrix is a complex matrix; one whose entries are complex numbers.
		In the study of complex vector spaces, an interesting class of matrices are the unitary ones; those whose inverse is the adjoint.
		An orthogonal matrix represents a rotation of given orthonormal axes (possibly with a reflection); a unitary matrix could be considered for this reason, as a complex rotation.
	www.maths.abdn.ac.uk /~igc/tch/eg1006/notes/node123.html (466 words)

	Welcome to this boffo site!
		In practice, the matrices in the constellation are special unitary matices.
		A special unitary matrix is a matrix of determinant 1 whose conjugate transpose is its inverse.
		Given a constellation {A1,..., Ak} of special unitary matrices, we define the diversity product of the constellation to be the minimum of the values det(Ai - Aj)^(1/n), i and j distinct.
	dimax.rutgers.edu /~rvenzke (987 words)

	Special unitary group - Enpsychlopedia (Site not responding. Last check: 2007-11-06)
		In mathematics, the special unitary group of degree $n is the group of n by n$ unitary matrices with determinant $1 and entries from the field \mathbb{C} of complex numbers, with the group operation that of$ matrix multiplication.
		This is a subgroup of the unitary group $\mathrm{U}(n), itself a subgroup of the general linear group \mathrm{GL}(n, \mathbb{C}) .$
		The special unitary group $\mathrm{SU}(n) is a real Lie group of dimension n^2 - 1 .$
	www.grohol.com /wiki/SU(2) (809 words)

	[No title]
		A matrix with one row is called a row vector, and a matrix with one column is called a column vector.
		e.g.: ó¨Matrix scalar multiplication¨L Multiplication of a matrix or a vector by a scalar is also straightforward:ó¨Transpose of a matrix¨0 Taking the transpose of a matrix is similar to that of a vector: The diagonal elements in the matrix are unaffected, but the other elements are switched.
		To find the rank of a matrix by hand, use Gauss elimination and the linearly dependant row vectors will fall out, leaving only the linearly independent vectors, the number of which is the rank.
	www.eng.fsu.edu /~palanki/orient/lec1.ppt (1262 words)

	Matrix Diagonalization
		is a Jordan block or Jordan box, and the matrix J is called a Jordan canonical form or Jordan normal form of the matrix A.
		The diagonal elements of the diagonal matrix D similar to the matrix A are the eigenvalues of the matrix A.
		In general case of the diagonalization of a matrix one must confine himself to the Jordan normal form (1).
	www.cs.ut.ee /~toomas_l/linalg/lin2/node19.html (281 words)

	Representation of Orthogonal or Unitary Matrices (Site not responding. Last check: 2007-11-06)
		An elementary reflector (or elementary Householder matrix) H of order n is a unitary matrix of the form
		Thus a complex H is not Hermitian (as it is in other representations), but it is unitary, which is the important property.
		This is useful, for example, when reducing a complex Hermitian matrix to real symmetric tridiagonal form, or a complex rectangular matrix to real bidiagonal form.
	www.netlib.org /lapack/lug/node128.html (250 words)

	Standard Matrix Groups
		Construct the special unitary group SU(n, K) corresponding to the n-dimensional vector space V over the field K = GF(q^2), where n >= 2 and q is a prime power.
		Given a matrix group G over the finite field K and a subfield S of K, construct the semilinear extension of G over the subfield S.
		Given a matrix group G and a permutation group H, construct action of the wreath product on the tensor power of G by H, which is the (image of) the wreath product in its action on the tensor power (of the space that G acts on).
	www.math.lsu.edu /magma/text296.htm (1648 words)

	sciforums.com - matrix operations
		I was never taught matrix operations past using determinants to solve differential equations.
		I'm still new to group theory typr stuff, but the SUxSU(2)xSU(3) looks like some combination of three special unitary groups, one in 1-D, one in 2-D, and one in 3-D. Hey, it makes perfect sense to me. I think I even used to know what those groups were, though I don't remember now.
		if the matrix is equal to the inverse of its hermitian adjoint, then it is a unitary matrix.
	www.sciforums.com /printthread.php?t=28690 (933 words)

	Matrix Theory: A Second Course
		They know Gauss-Jordan elimination and understand what a singular matrix is. They want a book that reviews the basics, but opens doors to more advanced topics that elementary books touch only lightly or not at all.
		It addresses the question of how to change a coordinate system in order to simplify a matrix—say to diagonalize the matrix, as is done to covariance matrices by principal component analysis.
		The last chapter covers a variety of special topics, including nonnegative matrices, higher-order eigenvalue problems, matrix equations and special matrices, including positive-definite matrices.
	www.riskbook.com /titles/ortega_j_(1987).htm (270 words)

	[No title] (Site not responding. Last check: 2007-11-06)
		These one-loop matrix elements can also be used as input in the determination of the corresponding next-to-leading order cross sections for heavy flavor photoproduction and in photon-photon reactions, as well as for any of the relevant crossed processes.
		Specializing to the five dimensional case, we show that these terms may result in gross distortions of the Kaluza-Klein gauge field masses, wave functions, and couplings to brane and bulk matter.
		The relevant matrix elements are computed by a direct numerical integration, in coordinate space, of the convolution of the wave function of the $B$ meson at rest and the wave function of the $X_c$ meson, boosted according with its recoil factor.
	www.physics.ohio-state.edu /~gee/KK/abstracts/0207ph.txt (14774 words)

	HW4 (Site not responding. Last check: 2007-11-06)
		Matrix Eigenvalues Equation, Eigenvectors, Application to normal modes calculations, Eigenvalues and Eigenvectors of Hermitian matrix, Rotation Matrices revisit.
		Verify that the eigenvalues are real and the eigenvectors are orthogonal to each other, the trace of sigma2 equal to the sum of eigenvalues, the product of eigenvlues equal to the determinant of sigma2.
		Rotation matrix: Refer to Q. 3 in H.W. Show that the net rotation matrix in that problem can also be expressed by Eq.
	www2.hawaii.edu /~plam/ph400/HW/HW4.html (202 words)

	Mathematical Methods (10/24.539) Overview of Linear Algebra Some Special Matrices
		Note that hermitian, skew-hermitian, and unitary matrices are, in general, complex matrices.
		The eigenvalues of a unitary matrix (or real orthogonal) matrix have absolute value of unity.
		Also note that the magnitude of each eigenvalue for the unitary matrix is indeed unity.
	gershwin.ens.fr /vdaniel/Doc-Locale/Cours-Mirrored/Methodes-Maths/white/math/s3/s3spm/s3spm.html (396 words)

	[No title] (Site not responding. Last check: 2007-11-06)
		The Hermite ensembles, for special values of $\beta$ ($1,2,4$) are better known in the literature as the Gaussian (Orthogonal, Unitary, Symplectic) Ensembles, while the Laguerre are known as the Wishart (real, complex, quaternion) matrices.
		The general $\beta>0$ case in the joint eigenvalue distributions has been studied as a theoretical model, with the interpretation of $\beta$ as an inverse temperature.
		In 2001 we have found matrix models models for general ($\beta>0$) Hermite and Laguerre ensembles, which we have used to prove central limit theorems for $\beta$ growing to infinity (the temperature dropping to a "freezing point").
	www.stat.berkeley.edu /~revelle/abs/dumitriu.txt (174 words)

	The Diagonal Torus of a Matrix Under Special Unitary Equivalence
		The Diagonal Torus of a Matrix Under Special Unitary Equivalence
		The Diagonal Torus of a Matrix Under Special Unitary Equivalence:SIAM Journal on Matrix Analysis and Applications Vol.
		Given a matrix $A$, a sufficient condition is given for the vector of diagonal elements of $UAV$ to cover a torus with specified base circle radii as $U$ and $V$ run over the special unitary group.
	epubs.siam.org /sam-bin/dbq/article/22651 (91 words)

	schur (MATLAB Function Reference) (Site not responding. Last check: 2007-11-06)
		The complex Schur form of a matrix is upper triangular with the eigenvalues of the matrix on the diagonal.
		The fact that the off-diagonal elements are so large indicates that this matrix has poorly conditioned eigenvalues; small changes in the matrix elements produce relatively large changes in its eigenvalues.
		They have been modified for complex problems and to handle the special case B = I.
	www.haystack.edu /computing/matlab5.3.1/help/techdoc/ref/schur.html (256 words)

	[No title] (Site not responding. Last check: 2007-11-06)
		A Hessenberg matrix is zero below the first subdiagonal.
		If the matrix is symmetric or Hermitian, the form is tridiagonal.
		This matrix has the same eigenvalues as the original, but more computation is needed to reveal them.
	www.nd.edu /~dtl/cheg258/cheg258-2000/cheg258-1999/cheg258-1998/YESTRDAY/cheg258-1997/cheg258-1996/notes/doc/hess.html (179 words)

	12 (Site not responding. Last check: 2007-11-06)
		We introduced some special matrices; then introduced eigenvalues and eigenvectors, and how to algebraically find them; and their relevance for quantum and classical physics.
		Noted the special fact about symmetric real and Hermitian matrices: they have a complete set of eigenvectors which can be chosen orthonormal, with real eigenvalues.
		We derived the secular equation for calculating a matrix' eigenvalues.
	www.emory.edu /PHYSICS/Faculty/Benson/320/notes/12/12.html (250 words)

	[No title] (Site not responding. Last check: 2007-11-06)
		% B = BANDRED(A, KL, KU) is a matrix unitarily equivalent to A % with lower bandwidth KL and upper bandwidth KU % (i.e.
		This code is a little wasteful in that % it computes certain elements which are immediately set to zero!
		In that case use the SVD!') end % Check for special case where order of left/right transformations matters.
	www.cs.unc.edu /~smp/COMP205/Program/Matrix/bandred.m (170 words)

	unitary - OneLook Dictionary Search
		Unitary : Online Plain Text English Dictionary [home, info]
		Phrases that include unitary: unitary transformation, unitary matrix, unitary system, special unitary group, unitary authority, more...
		Words similar to unitary: one, unitarily, whole, more...
	www.onelook.com /?w=unitary&ls=a (271 words)

	QR Method (Site not responding. Last check: 2007-11-06)
		The QR method can be used for arbitrary matrices, but becomes too laborious and instead it is used on special matrices, preferably Hessenberg or symmetric band matrices.
		This iteration converges for the non-symmetric case to an upper block triangular matrix, which has single elements or
		For the symmetric case, the limit matrix is a diagonal matrix.
	www.math.vt.edu /people/renardym/class_home/nova/bifs/node52.html (78 words)

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