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Topic: Hermitian matrix

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Symmetric matrix

Inverse matrix

Hermitian conjugate

Normal matrix

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Unitary matrix

Orthogonal matrix

Stochastic matrix

Hermitian

Matrix inversion

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	PlanetMath: Hermitian matrix
		The complex conjugate of a Hermitian matrix is a Hermitian matrix.
		Hermitian matrices is a vector subspace in the vector space of complex matrices.
		This is version 13 of Hermitian matrix, born on 2002-01-21, modified 2006-09-22.
	planetmath.org /encyclopedia/HermitianMatrix.html (197 words)

	PlanetMath: skew-Hermitian matrix
		The trace of a skew-Hermitian matrix is imaginary.
		The eigenvalues of a skew-Hermitian matrix are imaginary.
		This is version 18 of skew-Hermitian matrix, born on 2003-04-29, modified 2006-06-14.
	planetmath.org /encyclopedia/SkewHermitianMatrix.html (144 words)

	Hermitian and Skew Hermitian
		The conjugate of a complex matrix is the conjugate of all its entries.
		The tranjugate is the transpose of the conjugate.
		Let m be a hermitian matrix and consider x*m.y, where x is a row vector and y is a column vector.
	www.mathreference.com /la-sim,herm.html (825 words)

	Matrix Fundamentals
		A matrix is denoted by a capital letter; its elements are denoted by the same lower case letter with row/column subscripts.
		The elements in a square matrix that have same row index and column index are called the diagonal elements of the matrix, or simply the diagonal of the matrix.
		The transpose and Hermitian transpose of a matrix A are denoted by A
	www.intel.com /software/products/mkl/docs/WebHelp/mthapa4.html (579 words)

	Eigenvalue, eigenvector and eigenspace - Wikipedia, the free encyclopedia
		This is the characteristic polynomial of A: the eigenvalues of a matrix are the zeros of its characteristic polynomial.
		The operator norm of a square matrix is an upper bound for the moduli of its eigenvalues, and thus also for its spectral radius.
		In factor analysis, the eigenvectors of a covariance matrix correspond to factors, and eigenvalues to factor loadings.
	en.wikipedia.org /wiki/Eigenvector (4532 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 (4464 words)

	3.3 Matrix Precision and Shape
		A symmetric matrix is a square matrix with the property that the matrix is equal to its transpose.
		The complex analog of a symmetric matrix is a Hermitian matrix.
		Hermitian matrices are used in the same kinds of situations as symmetric matrices.
	www.roguewave.com /support/docs/hppdocs/lpkug/3-3.html (659 words)

	ASIC Design for Signal Processing
		For the matrix squaring, that means that the first location read for each output row contains both the data for the row to be read, and the column required for the first calculation.
		The technique described in this section allows for the calculation of any of the previous cases of normal matrix multiplication, in 16 cycles, when all of the source matrices are known to be Hermitian.
		When one, or none of the source matrices are Hermitian, then the techniques described for generic matrices may be used.
	www.geoffknagge.com /fyp/hermitian.shtml (631 words)

	Math Forum - Ask Dr. Math (Site not responding. Last check: 2007-09-17)
		A matrix is symmetric if A = A(t) where A(t) is the transpose of A. Such a matrix is necessarily square and has the leading diagonal as a line of symmetry.
		A matrix is skew-symmetric if A = -A(t) Such a matrix is also square, but the elements of the leading diagonal must all be zero.
		Since the eigenvalues of a real symmetric matrix are always real, the corresponding results apply to Hermitian matrices, namely: (a) The eigenvalues of a Hermitian matrix are real.
	www.mathforum.org /library/drmath/view/51963.html (515 words)

	Hermitian Matrices
		A Hermitian matrix is defined as a matrix that is equal to its Hermitian conjugate.
		Hermitian matrices have found an important application in modern physics, as the representations of measurement operators in Heisenberg's version of quantum mechanics.
		To each observable parameter of a physical system there corresponds a Hermitian matrix whose eigenvalues are the possible values that can result from a measurement of that parameter, and whose eigenvectors are the corresponding states of the system following a measurement.
	www.mathpages.com /home/kmath306/kmath306.htm (744 words)

	Matrix Manual: Special Matrices (Site not responding. Last check: 2007-09-17)
		matrix, A, is an nn Toeplitz matrix* in which a(i,j) is a function of {(i-j) modulo n}.
		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.toronto.edu /matrix/special.html (2192 words)

	Example 5: A Hermitian matrix (Site not responding. Last check: 2007-09-17)
		=> A is Hermitian (the ij-element is conjugate to the ji-element).
		=> B is skew Hermitian (a skew Hermite matrix).
		The diagonal elements of a skew Hermitian matrix are pure imaginary or zero.
	www.ee.oulu.fi /~mpa/matreng/eem1_3-5.htm (42 words)

	HB Files - The Harwell Boeing Sparse Matrix File Format (Site not responding. Last check: 2007-09-17)
		If the matrix is read from the file into memory, it is common to use the same compressed column storage to represent the matrix in memory.
		If the matrix is Hermitian, (which implies that it is square) (and which typically only occurs for complex arithmetic), then half of the nonzero off-diagonal elements don't need to be stored in the file.
		A indicates that the matrix is "assembled" (the typical case) while E indicates that the matrix is a finite element matrix that is going to be described as the "sum" of a set of smaller matrices.
	www.csit.fsu.edu /~burkardt/data/hb/hb.html (1251 words)

	Learn more about Riemann hypothesis in the online encyclopedia. (Site not responding. Last check: 2007-09-17)
		Visiting at the Institute for Advanced Study in 1972, he showed this result to Freeman Dyson, one of the founders of the theory of random matrices, which is of importance in physics due to the fact that the eigenstates of a Hamiltonian, for example the energy levels of an atomic nucleus, satisfy such statistics.
		Dyson saw that the statistical distribution found by Montgomery was exactly the same as the pair correlation distribution for the eigenvalues of a random Hermitian matrix.
		Subsequent work has strongly born out this discovery, and the distribution of the zeros of the Riemann zeta function is now believed to satisfy the same statistics as the eigenvalues of a random Hermitian matrix, the statistics of the so-called Gaussian Unitary Ensemble.
	www.onlineencyclopedia.org /r/ri/riemann_hypothesis.html (685 words)

	lapack/complex
		gams d2d1b file cppsvx.f prec complex for Solves a Hermitian positive definite system of linear, equations AX=B, where A is held in packed storage, and provides, an estimate of the condition number and error bounds on the, solution.
		file cgeesx.f prec complex for Computes the eigenvalues and Schur factorization of a general, matrix, orders the factorization so that selected eigenvalues, are at the top left of the Schur form, and computes reciprocal, condition numbers for the average of the selected eigenvalues,, and for the associated right invariant subspace.
		gams d4b4 file ctgexc.f prec complex for Reorders the generalized Schur decomposition of a complex, matrix pair (A,B) using a unitary equivalence transformation, so that the diagonal block of (A,B) with row index IFST is moved, to row ILST.
	www.netlib.org /lapack/complex (4342 words)

	Matrix Market: Glossary
		A defective matrix cannot be transformed to a diagonal matrix using similarity transformations.
		A matrix is diagonally dominant if the absolute value of each diagonal element is greater than the sum of the absolute values of the other elements in its row (or column).
		A Hessenberg matrix is `almost' triangular, that is, it is (upper or lower) triangular with one additional off-diagonal band (immediately adjacent to the main diagonal).
	math.nist.gov /MatrixMarket/glossary.html (756 words)

	Guide and Reference
		This matrix is used in Example 4.1 in referenced text [61].
		This matrix is used in Example 4.2 in referenced text [61].
		This matrix is used in Example 6.1 in referenced text [61].
	www.navo.hpc.mil /usersupport/IBM/ESSL/essl369.html (1506 words)

	Non-Hermitian random matrix models (Site not responding. Last check: 2007-09-17)
		Hermitian random matrices have been very successfully used to model Hamiltonian operators of closed quantum systems.
		However, random matrices are also of great importance in many other areas of physics in which they are not constrained to be Hermitian; these include: dynamics of neural networks, quantum mechanics of open systems, classical diffusion in random media and population biology.
		It is thus necessary to quantify statistical properties of eigenvectors of random non-Hermitian matrices.
	fy.chalmers.se /~frtbm/statphys_researchC.html (144 words)

	CenterSpace API Documentation - CenterSpace.NMath.Matrix
		A banded matrix is a matrix that has all its non-zero entries near the diagonal.
		A lower triangular matrix is a square matrix with all elements above the main diagonal equal to zero.
		A tridiagonal matrix is a matrix which has all its non-zero entries on the main diagonal, the super diagonal, and the subdiagonal.
	www.centerspace.net /doc/NMath/Matrix/ref (1869 words)

	Real/Imaginary Eigen Values
		We already know that a hermitian matrix is normal, and has orthogonal eigen vectors, but let's prove this for general hermitian operators, even those that cannot be implemented by a matrix.
		Thus m inverse is hermitian and real, and the inverse of a symmetric matrix is symmetric.
		Such a matrix has n independent eigen vectors, and their eigen values are all pure imaginary.
	www.mathreference.com /la-sim,realim.html (803 words)

	Types of Matrices (Site not responding. Last check: 2007-09-17)
		Positive Definite Matrices: A special sort of Hermitian matrix in which all the eigenvalues are positive.
		An example is the dynamical matrix describing vibrations of the atoms of a molecule or crystal, where
		A matrix whose columns are the eigenvectors of a Hermitian matrix is unitary; the unitarity is a consequence of the orthogonality of the eigenvectors.
	www.sst.ph.ic.ac.uk /people/a.mackinnon/Lectures/compphys/node56.html (447 words)

	Guide and Reference
		The imaginary parts of the diagonal elements of the complex Hermitian matrix A are assumed to be zero, so you do not have to set these values.
		On input, the imaginary parts of the diagonal elements of the complex Hermitian matrix A are assumed to be zero, so you do not have to set these values.
		On output, the imaginary parts of the diagonal elements of the matrix are set to zero.
	www.navo.hpc.mil /usersupport/IBM/PESSL/pessl249.html (983 words)

	[No title]
		ZHEMM performs one of the matrix-matrix operations or C := alphaBA + betaC, where alpha and beta are scalars, A is an hermitian* matrix and B and C are m by n matrices.
		On entry, SIDE specifies whether the hermitian matrix A appears on the left or right in the operation as follows: SIDE = 'L' or 'l' C := alphaAB + betaC, SIDE = 'R' or 'r' C := alphaBA + betaC, Unchanged on exit.
		Before entry, the leading m by n part of the array C must contain the matrix C, except when beta is zero, in which case C need not be set on entry.
	www.ibiblio.org /gferg/ldp/man/manl/zhemm.l.html (429 words)

	Sparse Linear Algebra - Untitled
		Octave includes a poly-morphic solver for sparse matrices, where the exact solver used to factorize the matrix, depends on the properties of the sparse matrix itself.
		The cost of determining the matrix type is small relative to the cost of factorizing the matrix itself, but in any case the matrix type is cached once it is calculated, so that it is not re-determined each time it is used in a linear equation.
		The band density is defined as the number of non-zero values in the matrix divided by the number of non-zero values in the matrix.
	www.gnu.org /software/octave/doc/interpreter/Sparse-Linear-Algebra.html (535 words)

	Matrix Theory
		Matrix Theory - Show that each matrix type is normal.
		Matrix Theory - (A)Show that if A is Hermitian, then iA is skew-Hermitian.
		Linear Algebra - Hermitian Similar Matrices - Suppose A & B are Hermitian matrices and AB=BA, show that A and B are simultaneously diagonalizable, ie, there exists an unitary matrix C so that both CAC adn CBC are diagonal.
	www.brainmass.com /homework-help/math/other/2949 (191 words)

	33.
		It is not straight forward to see that such a construction is also possible in the general case when not all of the characteristic roots of H differ, because it is not known, whether there are as before n linearly independent eigenvectors of H available.
		Since there exist for every characteristic root as many linearly independent eigenvectors as is determined by their multiplicity, the Hermitian matrix H has in each case n linearly independent eigenvectors.
		A unitary matrix U with these eigenvectors as columns yields the principal axes transformation for H, as can be confirmed as above in the case of all different characteristic roots.
	kr.cs.ait.ac.th /~radok/math/mat3/m133.htm (987 words)

	VII Hermitian and quadratic forms (Site not responding. Last check: 2007-09-17)
		For arbitrary real or complex values of the variables, the value of a Hermitian form is always real.
		One of the main tasks of this part is the transformation of Hermitian forms to a much simpler form by introduction of new variables.
		First of all, we will see how a Hermitian form behaves when new variables are introduced by linear substitution.
	kr.cs.ait.ac.th /~radok/math/mat3/m131.htm (242 words)

	Symmetric Eigenproblems
		The real symmetric or complex Hermitian matrix A is reduced to real tridiagonal form T.
		The routine xORGTR (or in the complex case xUNMTR) is provided to form Q explicitly; this is needed in particular before calling xSTEQR to compute all the eigenvectors of A by the QR algorithm.
		The routine xORMTR (or in the complex case xUNMTR) is provided to multiply another matrix by Q without forming Q explicitly; this can be used to transform eigenvectors of T computed by xSTEIN, back to eigenvectors of A.
	www.netlib.org /lapack/lug/node48.html (802 words)

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