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	Diagonal matrix - Wikipedia, the free encyclopedia
		In linear algebra, a diagonal matrix is a square matrix in which the entries outside the main diagonal are all zero.
		A diagonal matrix with all its main diagonal entries equal is a scalar matrix, that is, a scalar multiple λI of the identity matrix I.
		Because of the simple description of the matrix operation and eigenvalues/eigenvectors given above, it is always desirable to represent a given matrix or linear map by a diagonal matrix.
	en.wikipedia.org /wiki/Diagonal_matrix (482 words)

	Learn more about List of matrices in the online encyclopedia. (Site not responding. Last check: 2007-10-08)
		Diagonalizable matrix - A matrix similar to a diagonal matrix.
		Unitary matrix - A matrix whose conjugate transpose is its inverse.
		Permutation matrix - Matrix representation of a permutation.
	www.onlineencyclopedia.org /l/li/list_of_matrices.html (294 words)

	Diagonalizable matrix - Wikipedia, the free encyclopedia
		Diagonalizable matrices and maps are of interest because diagonal matrices are especially easy to handle: their eigenvalues and eigenvectors are known and one can raise a diagonal matrix to a power by simply raising the diagonal entries to that same power.
		An n-by-n matrix A over the field F is diagonalizable if and only if the sum of the dimensions of its eigenspaces is equal to n, which is the case if and only if there exists a basis of F
		One can also say that the diagonalizable matrices form a dense subset with respect to the Zariski topology: the complement lies inside the set where the discriminant of the characteristic polynomial vanishes, which is a hypersurface.
	en.wikipedia.org /wiki/Diagonalizable_matrix (793 words)

	Matrix exponential - Wikipedia, the free encyclopedia
		In mathematics, the matrix exponential is a function on square matrices analogous to the ordinary exponential function.
		Note that the exponential of a matrix is always a non-singular matrix.
		In fact, this map is surjective which means that every non-singular matrix can be written as the exponential of some other matrix (for this, it is essential to consider the field C of complex numbers and not R).
	en.wikipedia.org /wiki/Matrix_exponential (893 words)

	PlanetMath: similar matrix
		The canonical example is that a diagonalizable matrix
		This is version 8 of similar matrix, born on 2002-02-20, modified 2005-06-10.
		I already saw an entry which can't be seen in html mode ('triangular matrix').
	planetmath.org /encyclopedia/SimilarMatrix.html (250 words)

	Diagonalizable (Site not responding. Last check: 2007-10-08)
		Therefore every diagonalizable matrix (which is similar to a diagonal matrix) has n independent eigen vectors.
		Therefore a matrix is diagonalizable iff it has n independent eigen vectors.
		It is not similar to a diagonal matrix, i.e.
	www.mathreference.com /la-sim,diag.html (240 words)

	Glossary of Linear Algebra Terms
		The column space of a matrix is the subspace spanned by the columns of the matrix considered as vectors.
		The geometric multiplicity of an eigenvalue c of a matrix A is the dimension of the eigenspace of c.
		The row space of a matrix is the subspace spanned by the rows of the matrix considered as vectors.
	www.caam.rice.edu /~zhang/caam310/glossary.html (1005 words)

	Math 310 - Glossary of Linear Algebra Terms
		The column space of a matrix is the subspace spanned by the columns of the matrix considered as a set of vectors.
		An elementary matrix is a matrix that is obtained by performing an elementary row operation on an identity matrix.
		The row space of a matrix is the subspace spanned by the rows of the matrix considered as a set of vectors.
	www.math.uic.edu /math310/glossary.html (1060 words)

	ee_hw10
		, to diagonalize a matrix it may be necessary to work in a larger field.
		matrix, compare the number of multiplications involved in using the permutation definition of a determinant, versus the number involved in finding the determinant using elementary row operations.
		A van der Monde determinant is the determinant of a van der Monde matrix.
	www.math.ucla.edu /~baker/115ah.1.01f/handouts/ee_hw10/node1.html (667 words)

	PlanetMath: matrix exponential (Site not responding. Last check: 2007-10-08)
		is a matrix of the same type as
		Cross-references: inverse, determinant, trace, type, invertible, identity matrix, zero matrix, properties, satisfies, square, formula, diagonal matrix, diagonalizable, natural number, nilpotent, matrix, converges, dimension, square matrix, real, exponential
		This is version 2 of matrix exponential, born on 2003-04-06, modified 2004-04-21.
	planetmath.org /encyclopedia/MatrixExponential.html (112 words)

	Homework 11 Solutions (Site not responding. Last check: 2007-10-08)
		#6: The matrix A is similar to the matrix D in the middle of the factorization, so they have the same eigenvalues with the same multiplicities.
		Since D is a diagonal matrix, the eigenvalues are just the diagonal elements of D, so they are 5 (with multiplicity 2) and 4.
		Since we have a 2x2 matrix with 2 distinct eigenvalues, the matrix is diagonalizable, as there must be two linearly independent eigenvectors.
	www.evergreen.loyola.edu /~mpknapp/teaching/235-s01/HW11.html (852 words)

	[No title] (Site not responding. Last check: 2007-10-08)
		A matrix B in Mn is said to be similar to a matrix A in Mn if there exists a nonsingular matrix S in Mn such that B = S^(-1) A S. Observation.
		Theorem 1.3.7 An nxn matrix is diagonalizable if and only if there is a set of n linearly independent eigen vectors.
		Lemma 1.3.10 A block diagonal matrix is diagonalizable iff the blocks are diagonalizable.
	orion.math.iastate.edu /hentzel/class.510/May.28 (297 words)

	Linear Algebra Glossary (Site not responding. Last check: 2007-10-08)
		A border banded matrix is a 2 by 2 block matrix comprising a (large) leading block which is a square banded matrix, two dense rectangular side strips, and a (small) trailing block which is a square dense matrix.
		The inverse of a (nonsingular) circulant matrix is a circulant matrix.
		The transpose of a circulant matrix is a circulant matrix.
	www.csit.fsu.edu /~burkardt/papers/linear_glossary.html (13644 words)

	Eigenvalues, Determinants and Diagonalization (Site not responding. Last check: 2007-10-08)
		Recall: In MATLAB the transpose of a matrix is denoted with an apostrophe; i.e.
		AP = D, where D is a diagonal matrix whose entries on the diagonal are the eigenvalues of A corresponding to the eigenvectors in the respective columns of P.
		Note that the converse to this statement is not necessarily true; i.e., if a matrix is diagonalizable, it is not necessarily true that all its eigenvalues are distinct.
	math.ucsd.edu /~math20f/Fall/Lab7F/Lab7F.html (1541 words)

	ADDITIONAL PROBLEMS
		This corresponds to the case of underdamping or light damping--the mass oscillates, with amplitude going to zero as time goes on.
		4 we presented a matrix associated with age distributions in a population.
		This means that in the stable age distribution, the ratio of the age groups 0 to 1, 1 to 2, and 2 to 3 years in 6:3:1.
	distance-ed.math.tamu.edu /Math640/chapter5/node13.html (302 words)

	Linear Algebra (Math 2318) - Eigenvalues and Eigenvectors - Diagonalization (Site not responding. Last check: 2007-10-08)
		Definition 1 Suppose that A is a square matrix and further suppose that there exists an invertible matrix P (of the same size as A of course) such that
		is a diagonal matrix. In such a case we call A diagonalizable and say that P diagonalizes A.
		To see that neither of these are diagonalizable simply go back to Example 6 in the previous section to see that neither matrix has 3 linearly independent eigenvectors. In both cases we have only two linearly independent eigenvectors and so neither matrix is diagonalizable.
	tutorial.math.lamar.edu /AllBrowsers/2318/Diagonalization.asp (727 words)

	Orthogonal Eigen Vectors (Site not responding. Last check: 2007-10-08)
		If d is a diagonal matrix, it admits n orthogonal eigen vectors, namely the basis vectors of n space.
		This is a rigid rotation, hence the eigen vectors of pdq remain orthogonal.
		This is also known as a normal matrix, as described in the previous section.
	www.mathreference.com /la-sim,ortheig.html (392 words)

	Properties of Complex Symmetric Matrices
		Complex symmetry is a purely algebraic property, and it has no effect on the spectrum of the matrix.
		A complex symmetric matrix may not even be diagonalizable.
		We remark that the eigendecomposition (7.91) is the suitable adaptation of the corresponding decomposition for Hermitian matrices.
	www.cs.utk.edu /~dongarra/etemplates/node263.html (114 words)

	No Title
		For each of the matrices in (a)-(d) determine whether or not the matrix is diagonalizable.
		In Exercises a)-d) determine whether the given matrix is diagonalizable.
		AP is an orthogonal diagonalization of the given symmetric matrix A.
	math.la.asu.edu /~tracogna/MAT342/ch6_rev (149 words)

	Linear transformation - matrices
		In most of the questions I've done, when Q was a diagonal matrix, one of the transition matrices had columns consisting of eigenvectors of what would be equivalent to the matrix Q. In this case, the formula for P^n (AQ^nA^-1) works simply because A and A^-1 are inverses.
		Written in that basis, the transformation is represented by a diagonal matrix with eigenvalues on the diagonal.
		That is a matrix with 0s below the diagonal, eigenvalues on the diagonal and, where there are eigenvalues of multiplicity greater than 1, "blocks" with 1 just above the diagonal.
	www.physicsforums.com /showthread.php?threadid=98800 (1229 words)

	r3su05
		be the diagonal matrix whose diagonal entries are the eigenvalues of
		Warning: In order to be an orthogonal matrix it is not enough for the columns (respectively rows) to be an orthogonal set; they must satisfy the stronger condition of being orthonormal.
		So, notice that while every orthogonally diagonalizable matrix is diagonalizable, not every diagonalizable matrix is orthogonally diagonalizable, since there are examples of diagonalizable matrices which are not symmetric.
	www.math.okstate.edu /~myersr/3013/review/r3su05/r3su05.html (1501 words)

	Math 307 §E1: Objectives for Final Exam (Site not responding. Last check: 2007-10-08)
		Transpose of a vector, transpose of a matrix.
		Find the eigenvalues of a matrix, and a basis for the eigenspace belonging to each eigenvalue.
		Use the relations among the four fundamental subspaces of a matrix.
	orion.math.iastate.edu /alex/307/fx_obj.html (238 words)

	Diagonalizable matrix - Wikipedia, the free encyclopedia
		if there exists an invertible matrix P such that P
		However, there are also matrices that are not diagonalizable, even if complex numbers are used.
		Indeed, C has one eigenvalue (namely zero) and this eigenvalue has algebraic multiplicity 2 and geometric multiplicity 1.
	en.wikipedia.org /wiki/Diagonalizable (793 words)

	Review of Chapters 5 and 6 and loose ends
		The rank of a matrix is equal to the dimension of the column space, which is also equal to the dimension of the row space.
		The matrix D is a diagonal matrix, and the entries on the diagonal are the corresponding eigenvalues of A.
		The first idea is that it is easy to calculate in the case where you repeatedly multiply a matrix times one of its eigenvectors.
	www.math.uiuc.edu /~johnmaki/ch6review.html (1579 words)

	[No title] (Site not responding. Last check: 2007-10-08)
		The eigenvalues of A. The eigenvectors of A. The matrix P such that EMBED Equation.3 , a diagonal matrix.
		If v is an eigenvector of an invertible matrix A, then cv is an eigenvector of A-1 for all nonzero scalars c.
		An n x n matrix has n distinct real eigenvalues if and only if it is diagonalizable.
	www.ccaurora.edu /mat255/test5.doc (446 words)

	Assignment 4 Solution (Site not responding. Last check: 2007-10-08)
		In each case the geometric and algebraic multiplicities are equal and A is diagonalizable.
		The matrix of eigenvectors and corresponding diagonal matrix of eigenvalues are
		Since the geometric multiplicity is less than the algebraic multiplicity, matrix A is defective.
	math.uc.edu /~kingjt/MatrixMethods/276_4f00sol/276_4f00sol.html (209 words)

	ss3
		This is a triangular matrix, so its eigenvalues lie on the diagonal.
		It is not symmetric, but this does not rule out being diagonalizable.
		is diagonalizable since it is already diagonal, but it is not invertible, since it has zero determinant.
	www.math.okstate.edu /~myersr/3013/oldexam/ss3/ss3.html (309 words)

	Eigenvalues and Determinants (Site not responding. Last check: 2007-10-08)
		Note: In MATLAB the transpose of a matrix is denoted with an apostrophe; i.e.
		Therefore, there is a possibilitythat a matrix may appear to have zero determinant and yet be invertible.
		MATLAB defines the matrix P which has theeigenvectors of B as its columns and the matrix D as a diagonalmatrix with the corresponding eigenvalues along thediagonal.
	math.ucsd.edu /~math20f/Spring/Lab7S/Lab7S.html (1255 words)

	Math 311-200 - Final Exam Review -- Fall 2003
		A linear transformation L:V->V is said to be diagonalizable if and only if there is a basis for V relative to which the matrix for L is diagonal.
		Be able to determine whether or not L is diagonalizable.
		For an eigenvalue problem, a matrix A is diagonalizable if and only if it is similar to a diagonal matrix D. That is, there is an invertible matrix S such that D = S
	www.math.tamu.edu /~francis.narcowich/m311/f03h/m311h_rf.html (666 words)

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