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# Topic: Frobenius norm

###### In the News (Fri 17 May 13)

 Matrix norm A matrix norm is a norm on the vector space of all real or complex m-by-n matrices. These norms are used to measure the "sizes" of matrices, and allow to talk about limits of sequences and infinite series of matrices. The Frobenius norm of A is defined as www.ebroadcast.com.au /lookup/encyclopedia/ma/Matrix_norm.html   (254 words)

 Ferdinand Georg Frobenius Summary Ferdinand Georg Frobenius (October 26, 1849 - August 3, 1917) was a German mathematician, best-known for his contributions to the theory of differential equations and to group theory. Frobenius was born in Charlottenburg, a suburb of Berlin, and was educated at the University of Berlin. Group theory was one of Frobenius' principal interests in the second half of his career. www.bookrags.com /Ferdinand_Georg_Frobenius   (597 words)

 PlanetMath: matrix p-norm   (Site not responding. Last check: 2007-10-11) norms are very easy to calculate for an arbitrary matrix: Cross-references: Frobenius matrix norm, estimates, inequalities, eigenvalue, square root, calculate, norms, vector, terms, matrix, matrix norms, class In A_\infinity norm, the bounds need correction, as has already been pointed some other users. planetmath.org /encyclopedia/MatrixPNorm.html   (160 words)

 Norms and Condition Numbers of a Matrix Verify that the Frobenius norm satisfies the conditions of the matrix norm. Using the third property of the norm and the homogeneity of multiplication of a vector by a matrix, we have norm, respectively, then the calculation of the 2-norm is more complicated. www.cs.ut.ee /~toomas_l/linalg/lin1/node18.html   (377 words)

 MATH2071: LAB #9: Norms, Errors and Condition Numbers   (Site not responding. Last check: 2007-10-11) From the definitions of norms and errors, we can now define the condition number of a matrix, which will give us an objective way of measuring how "bad" the Hilbert matrix is, and how many digits of accuracy we can expect when solving a particular linear system. The Frobenius matrix norm is not vector-bound to the L2 vector norm, but is compatible with it; the Frobenius norm is much easier to compute than the L2 matrix norm. The norms of the matrix and its inverse exert some limits on the relationship between the forward and backward errors. www.csit.fsu.edu /~burkardt/math2071/lab_09.html   (1806 words)

 DLANTR DLANTR returns the value of the one norm, or the Frobenius norm, or the infinity norm, or the element of largest absolute value of a trapezoidal or triangular matrix A. where norm1 denotes the one norm of a matrix (maximum column sum), normI denotes the infinity norm of a matrix (maximum row sum) and normF denotes the Frobenius norm of a matrix (square root of sum of squares). Note that max(abs(A(i,j))) is not a matrix norm. www.math.ucla.edu /computing/docindex/lapack-manpages-man-430.html   (356 words)

 PlanetMath: Frobenius matrix norm A nice property of the norm is that See Also: matrix norm, matrix p-norm, vector norm, vector p-norm, Schur's inequality, trace, transpose, transpose, matrix logarithm This is version 14 of Frobenius matrix norm, born on 2001-10-06, modified 2006-10-04. planetmath.org /encyclopedia/MatrixFNorm.html   (130 words)

 Norm Method (Default) Calculates the Frobenius norm of the matrix, the square root of the sum of the absolute squares of matrix elements. Calculates the Euclidean norm of the matrix, the largest singular value of the matrix. One Norm = 26 Frobenious = 23.3238075793812 Euclidean = 22.7204825366697 Infinity = 26 www.bluebit.gr /matrix/version_31/Norm.htm   (168 words)

 MATH2071: LAB #5: Norms, Errors and Condition Numbers   (Site not responding. Last check: 2007-10-11) The spectral matrix norm is not vector-bound to any vector norm, but it almost" is. This norm is useful because we often want to think about the behavior of a matrix as being determined by its largest eigenvalue, and it often is. But there is no vector norm for which it is always true that These quantities depend on the vector norm used, they cannot be defined in cases where the divisor is zero, and they are problematic when the divisor is small. The one norm or the inf norm are faster than the two norm. www.math.pitt.edu /~sussmanm/2071Spring05/lab05/index.html   (2953 words)

 Vector And Matrix Norms In all cases, the norm of a scalar is equal to its absolute value. The Frobenius norm of x is the square root of the sum of the squares of the absolute value of the elements of x. If x is not a vector or an empty matrix and p is not present or it is equal to 2, the return value is the maximum absolute singular value corresponding to the matrix x. www.omatrix.com /manual/normfunction.htm   (388 words)

 norm :: Functions (MATLAB Function Reference)   (Site not responding. Last check: 2007-10-11) The norm of a matrix is a scalar that gives some measure of the magnitude of the elements of the matrix. returns a different kind of norm, depending on the value of The infinity norm, or largest row sum of www.mathworks.com /access/helpdesk/help/techdoc/ref/norm.html   (141 words)

 frobenius norm The Frobenius norm is defined by an nxn matrix A by A_F=sum[(aij^2)^(1/2) i=1..n,j=1..n] I'm having trouble showing A+B i think hurkyl is assuming you meant the euclidean norm, and then your formula would simply be the norm of a vector in euclidean n space. the properties of this norm are probably based on some inequality they teach at the beginnig of many courses called the schwartz inequality (see chapter 0 or 1 of spivak's calculus book). www.physicsforums.com /showthread.php?t=32162   (306 words)

 ?langt Returns the value of the 1-norm, Frobenius norm, infinity-norm, or the largest absolute value of any element of a general tridiagonal matrix. The routine returns the value of the 1-norm, or the Frobenius norm, or the infinity norm, or the element of largest absolute value of a real/complex tridiagonal matrix where norm1 denotes the 1-norm of a matrix (maximum column sum), normI denotes the infinity norm of a matrix (maximum row sum) and normF denotes the Frobenius norm of a matrix (square root of sum of squares). www.intel.com /software/products/mkl/docs/WebHelp/lau/functn_langt.html   (98 words)

 17.2 ELL_Matrix Methods A matrix norm is any scalar-valued function on a matrix, denoted by the double bar notation A frequently used matrix norm is the Frobenius norm: The general class of matrix norms known as the p-norms are defined in terms of vector norms by: www.lanl.gov /Caesar/node289.html   (164 words)

 SALSA Analysis Modules: simple.c File Reference Compute and store the infinity norm of the antisymmetric part of the matrix. Compute and store the Frobenius norm of the antisymmetric part of the matrix. Compute and store the Frobenius norm of the symmetric part of the matrix. www.cs.utk.edu /~eijkhout/salsa-modules-docs/simple_8c.html   (686 words)

 SVD and LSI Tutorial 2: Computing Singular Values The Frobenius Norm of a matrix is defined as the square root of the sum of the absolute squares of its elements. Since a column vector is a one-column matrix and a row vector is a one-row matrix, the Frobenius Norm of these matrices equals the length (L) of the vectors. Thus, normalized unit vectors are vectors normalized in terms of their Frobenius Norm. www.miislita.com /information-retrieval-tutorial/svd-lsi-tutorial-2-computing-singular-values.html   (1199 words)

 Norm Method A NormType enumeration value that specifies the type of norm to be returned. The following example calculates and prints the four different types norms of a matrix using the Norm method. Frobenius norm of the matrix (square root of sum of squares). www.bluebit.gr /NET/Library/Matrix-Norm.html   (148 words)

 13.2 ELL_Matrix Class A vector of Overlapped Vectors that is used for matvecs. An estimate of the two norm of the ELL Matrix, taken to be the midpoint of the range. The possible range of the two norm of the ELL Matrix. www.lanl.gov /Caesar/node218.html   (527 words)

 How to Measure Errors In order to measure the error in vectors, we need to measure the size or norm of a vector x. A popular norm is the magnitude of the largest component, Errors in matrices may also be measured with norms. www.netlib.org /lapack/lug/node75.html   (674 words)

 Further Details: How to Measure Errors An error bound that uses a given norm may be changed into an error bound that uses another norm. This is accomplished by multiplying the first error bound by an appropriate function of the problem dimension. The two-norm, Frobenius norm, and singular values of a matrix do not change if the matrix is multiplied by a real orthogonal (or complex unitary) matrix. www.netlib.org /lapack/lug/node76.html   (499 words)

 osborne (Robust Control Toolbox) Compute an upper bound on the structured singular value via the Osborne method. Form the n by n matrix F whose elements are the largest singular values of the blocks of the matrix A. Compute the diagonal scaling D that minimizes the Frobenius norm of www.weizmann.ac.il /matlab/toolbox/robust/osborne.html   (271 words)

 lapack-z/zlansy.html   (Site not responding. Last check: 2007-10-11) ARGUMENTS NORM (input) CHARACTER*1 Specifies the value to be returned in ZLANSY as described above. UPLO (input) CHARACTER*1 Specifies whether the upper or lower triangular part of the symmetric matrix A is to be referenced. WORK (workspace) DOUBLE PRECISION array, dimension (LWORK), where LWORK >= N when NORM = 'I' or '1' or 'O'; oth- erwise, WORK is not referenced. www.math.utah.edu:8080 /software/lapack/lapack-z/zlansy.html   (193 words)

 Matrix Norms is called an operator norm or induced norm. The geometric interpretation of such a norm is that it is the maximum length of a unit vector after transformation by Furthermore, an operator norm is a matrix norm (i.e. www.cs.unc.edu /~dm/UNC/COMP205/LECTURES/LINALG/lec5/node4.html   (111 words)

 Transfer Residual Errors to Backward Errors. In fact, practical purposes will be served if we can determine upper bounds for the norms of these (nearly) optimal matrices. -backward stable for the pair with respect to the norm -backward stable for the triplet with respect to the norm www.cs.utk.edu /~dongarra/etemplates/node277.html   (227 words)

 dlange(l): return value of one norm, or ... - Linux man page DLANGE - return the value of the one norm, or the Frobenius norm, or the infinity norm, or the element of largest absolute value of a real matrix A DLANGE returns the value of the one norm, or the Frobenius norm, or the infinity norm, or the element of largest absolute value of a real matrix A. DLANGE returns the value DLANGE = (max(abs(A(i,j))), NORM = 'M' or 'm' ((norm1(A), NORM = '1', 'O' or 'o' ((normI(A), NORM = 'I' or 'i' ((normF(A), NORM = 'F', 'f', 'E' or 'e' www.die.net /doc/linux/man/manl/dlange.l.html   (213 words)

 slanhs(l): return value of one norm, or ... - Linux man page SLANHS - return the value of the one norm, or the Frobenius norm, or the infinity norm, or the element of largest absolute value of a Hessenberg matrix A SLANHS returns the value of the one norm, or the Frobenius norm, or the infinity norm, or the element of largest absolute value of a Hessenberg matrix A. SLANHS returns the value SLANHS = (max(abs(A(i,j))), NORM = 'M' or 'm' ((norm1(A), NORM = '1', 'O' or 'o' ((normI(A), NORM = 'I' or 'i' ((normF(A), NORM = 'F', 'f', 'E' or 'e' www.die.net /doc/linux/man/manl/slanhs.l.html   (204 words)

 The Singular Value Decomposition of an Image This is not the most common matrix norm, and not always very convenient, but it is related to the so-called L -norm) is related to the concept of energy and for this reason the Frobenius matrix norm is sometimes useful in image processing. Compute the relative error in the Frobenius norm for both of the approximation. amath.colorado.edu /courses/4720/2000Spr/Labs/SVD/svd.html   (1462 words)

 Preliminaries   (Site not responding. Last check: 2007-10-11) by evaluating the Frobenius norm of M. In general, M will be a complex matrix, and its norm will be difficult to evaluate analytically. The Frobenius norm of this matrix is easily calculated to equal 4. For a hot'' configuration, the unitary matrices making up M will be rotated away from their principle axes, and the norm of M will be slightly less than 4, so that www-astro.physics.ox.ac.uk /~rjohnson/thesis/node28.html   (414 words)

 Class Protein   (Site not responding. Last check: 2007-10-11) This procedure gets a rotation matrices array as input, and returns the frobenius norm of each matrix. This procedure gets an array of frobenius norms as input and returns all the sequences which the frobenious norm of each element in the sequence is less than the frobenius constant. This procedure Gets as input a specific shift and a protein B. It calculates the rotation matrices of each two unit-vectors (= 3 alpha-carbons), and returns an array of those rotation matrices. www.cs.bgu.ac.il /~klara/AhazGal/API/Protein.html   (1046 words)

 SALSA Analysis Modules: simple.c Source File   (Site not responding. Last check: 2007-10-11) 00035 This element and all following are sequential; see \ref options; 00036 - "symmetry-anorm" : infinity norm of anti-symmetric part; see SymmetryANorm(). 00037 - "symmetry-fsnorm" : Frobenius norm of symmetric part; see 00038 SymmetryFSNorm(). 00039 - "symmetry-fanorm" : Frobenius norm of anti-symmetric part; 00040 see SymmetryFANorm(). www.tacc.utexas.edu /~eijkhout/doc/anamod/html/simple_8c-source.html   (657 words)

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