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Weyl Groups A Cartan matrix is: 2 0 -1 0 0 0 0 2 0 -1 0 0 -1 0 2 -1 0 0 0 -1 -1 2 -1 0 0 0 0 -1 2 -1 0 0 0 0 -1 2 The Coxeter-Dynkin diagram is A Cartan matrix is: 2 0 -1 0 0 0 2 -1 0 0 -1 -1 2 -1 0 0 0 -1 2 -1 0 0 0 -1 2 The Coxeter-Dynkin diagram is A Cartan matrix is: 2 0 -1 0 2 -1 -1 -1 2 The Coxeter-Dynkin diagram is www.valdostamuseum.org /hamsmith/Weyl.html   (5287 words)

Linear Algebra The Wronskian matrix is created by putting each function as the first element of each column, and filling in the rest of each column by the (i-1)-th derivative, where i is the current row. In> A:={{4,-2,4,2},{-2,10,-2,-7},{4,-2,8,4},{2,-7,4,7}} Out> {{4,-2,4,2},{-2,10,-2,-7},{4,-2,8,4},{2,-7,4,7}}; In> R:=Cholesky(A); Out> {{2,-1,2,1},{0,3,0,-2},{0,0,2,1},{0,0,0,1}}; In> Transpose(R)*R = A Out> True; In> Cholesky(4*Identity(5)) Out> {{2,0,0,0,0},{0,2,0,0,0},{0,0,2,0,0},{0,0,0,2,0},{0,0,0,0,2}}; In> Cholesky(HilbertMatrix(3)) Out> {{1,1/2,1/3},{0,Sqrt(1/12),Sqrt(1/12)},{0,0,Sqrt(1/180)}}; In> Cholesky(ToeplitzMatrix({1,2,3})) In function "Check" : CommandLine(1) : "Cholesky: Matrix is not positive definite" The minor is the determinant of the matrix obtained from M by deleting the i-th row and the j-th column. yacas.sourceforge.net /refchapter9.html   (2078 words)

s06.txt Hence the matrix of L with respect to {1,x,x^2} is A = 0 0 2 0 1 0 0 0 2 b. Hence the matrix of L with respect to {1,x,1+x^2} is B = 0 0 0 0 1 0 0 0 2 c. Since S^{-1} is the transition matrix from [v1,v2] to [u1,u2], we may compute the matrix of L in the V basis as C=SBS^{-1}, using Theorem 4.3.1 on p.214. www.math.wustl.edu /~victor/classes/ma309/s06.txt   (2078 words)

Computer Science 121 - Summer 2002 - Lab Assigment 5 Exit the program ENTER AN OPTION:8 Here is matrix a: numRows= 3 numCols= 4 1.0 2.0 3.0 4.0 5.0 6.0 7.0 8.0 9.0 10.0 11.0 12.0 Here is matrix b: numRows= 3 numCols= 4 12.0 11.0 10.0 9.0 8.0 7.0 6.0 5.0 4.0 3.0 2.0 1.0 1. When this method is called the matrix that is passed to the method as a parameter will be added to the current instance and the (newly created) resulting matrix will be returned by the method. Generate and print the transpose of matrix a 6. www-unix.oit.umass.edu /~rcasstev/labs/lab5.html   (1212 words)

t1.txt The reduced row-echelon form of the matrix 1 1 0 1 is 1 0 0 1 0 1 0 0 1 1 1 1 1 0 0 1 [A] 0 1 0 0 [B] 1 0 0 1 [C] 0 1 0 0 [D] 0 1 0 1 0 1 2 3 4 2. The matrix equality BB-B=(B-I)B is correct [A] for 1x1 matrix B only [B] if B is a symmetric matrix [C] if B is a column matrix (or a column vector) [D] The correct answer is not given by [A],[B], or [C] 5. [B] Matrix [ A b ] is an augmented matrix of the linear system Ax=b. www-math.cudenver.edu /~aknyazev/teaching/94/3191/t1.txt   (1212 words)

LLLGCD algorithm: gcd of 5 integers The unimodular matrix is -1 0 -1 2 1 0 3 -1 -2 0 2 2 -3 2 -2 1 4 4 3 -3 2 -3 -2 -1 -1 b[5]=[2,-3,-2,-1,-1] is a multiplier of lengthsquared 19. Two other randomly found examples: gcd(103,500,1005,204,60) The unimodular matrix is 3 0 -1 4 -2 4 -2 0 2 3 4 1 0 -3 -5 -2 -5 2 4 -2 1 -3 2 -3 0 b[5]=[1,-3,2,-3,0] is a multiplier of lengthsquared 23. gcd(2,5,14,23,29) alpha = 1 The unimodular matrix found is 0 1 -2 1 0 2 -1 -1 1 0 -1 2 -1 -1 1 2 -1 -2 0 1 1 1 0 1 -1 Call the rows b[1],..,b[5]. www.numbertheory.org /lll/example5.html   (243 words)

matrix.c */ 1,0,0,5}; /* Note that this is a permutation matrix that switches columns according to * the pattern in the matrix. */ 0,0,0,1, /* Thus the matrix is TRANSPOSE of that */ 1,0,0,0, /* suggested by the formatting. */ 0,3,0,3, /* Thus the matrix is TRANSPOSE of that */ 0,0,4,0, /* suggested by the formatting. www.cs.arizona.edu /classes/cs433/fall04/matrix.c   (243 words)

constant.w += Matrix GetMatrix(GaloisElement g) { Vector component; Matrix aux(7,7); component=Vector(27,0,0L); component.vector[(LineInCubic)"L(0,0)"]=1; component.vector[(LineInCubic)"L(1,0)"]=1; component.vector[(LineInCubic)"L(1,2)"]=1; component=projection*(g*component); aux.matrix[0]=component; component=Vector(27,(LineInCubic)"L(0,0)",1); component=projection*(g*component); aux.matrix[1]=component; component=Vector(27,(LineInCubic)"L(1,0)",1); component=projection*(g*component); aux.matrix[2]=component; component=Vector(27,(LineInCubic)"L(2,0)",1); component=projection*(g*component); aux.matrix[3]=component; component=Vector(27,(LineInCubic)"M(0,0)",1); component=projection*(g*component); aux.matrix[4]=component; component=Vector(27,(LineInCubic)"M(1,0)",1); component=projection*(g*component); aux.matrix[5]=component; component=Vector(27,(LineInCubic)"M(2,0)",1); component=projection*(g*component); aux.matrix[6]=component; return(aux); } @ This function is public @ Then the generators of the Galois group appears as the kernel of the transposed matrix. += #ifdef __cplusplus maybeextern Matrix action_c,action_i,action_j,action_k; #endif @ We compute them using the action of the Galois group on the lines and the projection matrix. www.math.princeton.edu /~ytschink/bin/cubicsurf/constant.w   (243 words)

87521.011122&ELEMENT_SET=DECL The copper/refractory metal matrix composite of claim 1 wherein the refractory metal is selected from the group consisting of tungsten, molybdenum, chromium, iridium, osmium, tantalum, niobium, ruthenium, rhenium, rhodium, hafnium, zirconium and mixtures thereof. The copper/refractory metal matrix composite of claim 7 comprising: from about 10% to about 20% by weight copper; from about 0.3% to about 0.4% by weight cobalt; and, an amount of phosphor such that the P/Co weight ratio ranges from about 0.3 to about 0.3 8; and the remainder is tungsten. The process of claim 18 wherein the copper/refractory metal matrix composite comprises: from about 10% to about 20% by weight copper; from about 0.3% to about 0.4% by weight cobalt; and, an amount of phosphor such that the P/Co weight ratio ranges from about 0.3 to about 0.38; and the remainder is tungsten. www.wipo.int /cgi-pct/guest/getbykey5?KEY=01/87521.011122&ELEMENT_SET=DECL   (5976 words)

Citations: Error detecting and error correcting codes - Hamming (ResearchIndex) It can be given by the generator matrix G in standard form [I 4 where A = # # # # 0 1 1 1 0 1 1 1 0 # # #. Thus a check matrix will be # 1 0 1 1 1 0 0 1 1 0 1 0 1 0 1 1 1 0 0 0 1 #. ....equivalent if and only if there exist a non singular matrix M and a monomial matrix N such that MGN = G #, with isomorphism if N is a permutation matrix and equality if N = I n, n being the block length of the codes. citeseer.ist.psu.edu /context/19768/0   (5976 words)

eisBTfry00048.txt %F A093406 We use a 4 X 4 matrix corresponding to the characteristic polynomial (x - 1)^4 - 2 = 0 = x^4 - 4x^3 + 6x^2 - 4x - 1 = 0, being [0 1 0 0 / 0 0 1 0 / 0 0 0 1 / 1 4 -6 4]. Let the matrix = M. Perform M^n * [1,1,1,1]. %C A092781 This maximum number is achieved by the 'good' binary trees defined in the paper. www.research.att.com /~njas/sequences/eisBTfry00048.txt   (5976 words)

Linear Algebra Part One Given a matrix A, the inverse of the partitioned matrix (A, 0, 0, 1) (where 0 is a matrix each of whose elements is zero and 1 is an identity matrix), is the matrix (1/A, 0, 0, 1). Given a matrix A, the inverse of the matrix (A, 0, 0, 1), where 0 is a matrix each of whose elements is zero and 1 is an identity matrix, is the matrix (1/A, 0, 0, 1). Definition A matrix is said to be *skew-symmetric* iff its transpose is equal to the negative of the given matrix. www.rism.com /LinAlg/LAone.htm   (5976 words)

polynomial.mws The matrix Laurent polynomial W is de fined as " }{XPPEDIT 18 0 "diag(p[1]..p[n])" "6#-%%diagG6#;&%\"pG6#\" \"\"&F(6#%\"nG" }{TEXT -1 56 ". It is based on the generalization to the matrix case of sev eral theorems in \"Stationary Subdivision\" by Cavaretta, Dahmen and M icchelli [CDM], and to the multivariate case of several theorems fro m \"Matrix subdivision\" by Cohen, Dyn and Levin [CDL]. The argument W allows one to define a contraction function " } {XPPEDIT 18 0 "abs(W)[infinity] " "6#&-%$absG6#%\"WG6#%)infinityG" } {TEXT -1 147 ", similarly, WD defines the contraction function for th e difference scheme. mrl.nyu.edu /projects/subdivision/maple/polynomial.mws   (595 words)

791 Matrix functions -------------------- FFT over 800,000 random values______________________ (sec): 0.823 Eigenvalues of a 320x320 random matrix______________ (sec): 0.963 Determinant of a 650x650 random matrix______________ (sec): 0.8977 Cholesky decomposition of a 900x900 matrix__________ (sec): 0.503 Inverse of a 400x400 random matrix__________________ (sec): 0.6413 ------------------------------------------------------ Trimmed geom. Matrix functions > -------------------- > FFT over 800,000 random values______________________ (sec): 0.9717 > Eigenvalues of a 320x320 random matrix______________ (sec): 1.215 > Determinant of a 650x650 random matrix______________ (sec): 1.166 > Cholesky decomposition of a 900x900 matrix__________ (sec): 0.6697 > Inverse of a 400x400 random matrix__________________ (sec): 0.9513 > ------------------------------------------------------ > Trimmed geom. Programmation ------------------ 750,000 Fibonacci numbers calculation (vector calc)_ (sec): 1.662 Creation of a 2250x2250 Hilbert matrix (matrix calc) (sec): 2.217 Grand common divisors of 70,000 pairs (recursion)___ (sec): 0.9833 Creation of a 220x220 Toeplitz matrix (loops)_______ (sec): 2.975 Escoufier's method on a 37x37 matrix (mixed)________ (sec): 3.369 ------------------------------------------------------ Trimmed geom. www.octave.org /mailing-lists/octave-maintainers/2004/791   (586 words)

dodgson The 1-by-1 matrix that you get at the end is the answer (that is, the determinant of the original matrix). Here's a kludge we can use: replace the matrix by a t^0 t^1 t^3 t^6 t^0 b t^0 t^1 t^3 t^1 t^0 c t^0 t^1 t^3 t^1 t^0 d t^0 t^6 t^3 t^1 t^0 e (where the exponents of t are successive triangular numbers). Here I should digress and reveal that for all k between 1 and n, the k-by-k matrix obtained by Dodgson's algorithm is just the matrix consisting of all k^2 of the (n-k+1)-by-(n-k+1) "connected minors" of the original matrix. www.math.wisc.edu /~propp/somos/dodgson   (1213 words)

Creation of Matrices A matrix or a vector may also be created by coercing a sequence of ring elements into the appropriate parent matrix structure. Given a matrix A of any type, return the same matrix but having as parent the appropriate matrix algebra if A is square, or the appropriate R-matrix space otherwise. The sequence Q may be a sequence of m sequences, each of length n and having entries in a ring S, in which case the rows of the matrix are specified by the inner sequences. www.umich.edu /~gpcc/scs/magma/text742.htm   (1213 words)

ASPN : Python Cookbook : initialize a dict representing distance matrix between 2 list of strings ''' two list of names L1 =['a','b','c'] L2 = ['d','e','f','g'] D = len(L1) x len(L2) type distance 'matrix' intialized to 0 ''' dict(zip(L1,[dict(zip(L2,[0]*len(L2)))]*len(L1))) ''' output {'a': {'e': 0, 'f': 0}, 'c': {'e': 0, 'f': 0}, 'b': {'e': 0, 'f': 0}} ''' Title: initialize a dict representing distance matrix between 2 list of strings ASPN : Python Cookbook : initialize a dict representing distance matrix between 2 list of strings aspn.activestate.com /ASPN/Cookbook/Python/Recipe/162394   (1213 words)

capelli.tex If a linkable matrix has $m$ links $(i_1 In other words define the operators: $$ \displaylines{ D_i=h {\partial\over \partial p_{b_i,i}} {\partial\over \partial x_{b_i,i}}\qquad{\rm and}\qquad \Delta_i=x_{b_i,i}p_{b_i,i}{\partial\over\partial p_{b_i,i}} {\partial\over\partial x_{b_i,i}}.\cr \noalign{\hbox{Then let}} w(G,K)=\Bigl(\prod_{i\in K}D_i \prod_{i\in I\setminus K}\Delta_i\Bigr) w(K)\cr} $$ For instance, the matrix $$ G=\pmatrix{ 4&5&1&8&7&6&9&2&3\cr 2&8&2&1&8&8&8&8&2\cr 1&2&3&4&5&6&7&8&9\cr 0&0&0&0&0&1&0&0&0\cr } $$ has three links $(1,3)$, $(2,8)$ and $(3,9)$. Let $X=(x_{ij})$, $P=(p_{ij})$ $(1\le i,j\le n)$ be as before, but now they are symmetric matrices: $x_{i,j}=x_{j,i}$ and $p_{i,j}=p_{j,i}$, their entries satisfying the same commutation rules. www.math.temple.edu /~zeilberg/mamarim/mamarimTeX/capelli.tex   (1213 words)

An introduction to MATRICES An identity matrix I is a diagonal matrix with all diagonal element = 1. [7 0 0] [0 5 0] [0 0 6] The matrix is denoted by diag(7, 5, 6) A diagonal matrix is a square matrix with all de non-diagonal elements 0. home.scarlet.be /~ping1339/matr.htm   (921 words)

CSPICE Routines: XPOSEG_C For example suppose you have the following declarations SpiceDouble matrix [1003][800]; If the transpose of the matrix is needed, it may not be possible to fit a second matrix requiring the same storage into memory. matrix xposem m[0][0] m[0][0] m[0][1] m[1][0] m[0][2] m[2][0]. Transpose a matrix of arbitrary size (in place, the matrix need not be square). www.gps.caltech.edu /~marsdata/cspice/xposeg_c.html   (921 words)

An introduction to MATRICES An identity matrix I is a diagonal matrix with all diagonal element = 1. [7 0 0] [0 5 0] [0 0 6] The matrix is denoted by diag(7, 5, 6) A diagonal matrix is a square matrix with all de non-diagonal elements 0. www.ping.be /~ping1339/matr.htm   (921 words)

An introduction to MATRICES An identity matrix I is a diagonal matrix with all diagonal element = 1. [7 0 0] [0 5 0] [0 0 6] The matrix is denoted by diag(7, 5, 6) A diagonal matrix is a square matrix with all de non-diagonal elements 0. home.scarlet.be /~ping1339/matr.htm   (921 words)

Solutions to Problem Set #5 In either case, the principle minor determinants alternate negative and positive (-4<0 and 28>0 or –16<0 and 28>0), the matrix is positive definite, and the solution is a maximum of profits. Its first principle minor matrix determinant is 2 > 0, its second principle minor matrix determinant is 12-9=3 > 0, and its third principle minor determinant is its own determinant, which is 4 > 0. The equation is homogenous (ie, the vector of constants is the zero vector) and the determinant of the matrix is 4, which is not zero. www.union.edu /PUBLIC/ECODEPT/schmidsj/eco138/psol5.html   (938 words)

Rotation matrix from a vector? Then, build the matrix by inserting the vectors into the matrix columns: a'[0] b'[0] c'[0] t[0] a'[1] b'[1] c'[1] t[1] a'[2] b'[2] c'[2] t[2] 0 0 0 1 where _t_ is the translation you wish to apply after the rotation. A much more efficient way to it if the original axis _a_ is always the x axis is to build the matrix yourself. First, you need to find two vectors _b'_ and _c'_ which are perpendicular to _a'_. public.kitware.com /pipermail/vtkusers/2000-May/050398.html   (402 words)

An introduction to MATRICES An identity matrix I is a diagonal matrix with all diagonal element = 1. [7 0 0] [0 5 0] [0 0 6] The matrix is denoted by diag(7, 5, 6) A diagonal matrix is a square matrix with all de non-diagonal elements 0. www.ping.be /~ping1339/matr.htm   (402 words)

hw2.soln $ (b) Note: $I = A - B.$ (c) Let the subspace consist of all multiplies of the matrix $\left[ \matrix {0 & 1 \cr 0 & 0 \cr} \right].$ \smallskip \hrule \smallskip \item{\bf 2.} (a) Show that the set of all invertible matrices in ${\bf R}^{2 \times 2}$ is not a subspace. {\bf Answer:} (a) The smallest such subspace would be all scalar multiplies of the matrix $A,$ that is, the subspace $\left\{ \left[ \matrix {c & 0 \cr 0 & 0 \cr} \right] : c \in {\bf R} \right\}. {\bf Answer:} (a) The identity matrix $I$ is invertible, but $I - I =0$ is not invertible. www.mcs.drexel.edu /~rboyer/courses/math507/hw2.soln   (312 words)

An introduction to MATRICES An identity matrix I is a diagonal matrix with all diagonal element = 1. [7 0 0] [0 5 0] [0 0 6] The matrix is denoted by diag(7, 5, 6) A diagonal matrix is a square matrix with all de non-diagonal elements 0. home.scarlet.be /~ping1339/matr.htm   (921 words)

Matrix and Quaternion FAQ Given that each 4x4 rotation matrix is guaranteed to have 10 elements with value zero (0), 2 elements with value one (1) and four others of arbitary value, over 75% of every matrix operation is wasted. Then the determinant is calculated as follows: n --- \ i det M = / M * submat M * -1 --- 0,i 0,i i=1 where submat M defines the matrix composed of all rows and columns of M ij excluding row i and column j. Using linear interpolation, the interpolated rotation matrix is generated using a blending equation with the parameter T, which ranges from 0.0 to 1.0. www.flipcode.com /documents/matrfaq.html   (921 words)

Matrix Operations for Image Processing This matrix can also be used to complement the colors in an image by specifying a saturation value of -1.0. Then a matrix that rotates about the 1.0,1.0,1.0 diagonal can be constructed like this: One nice property of this saturation matrix is that the luminance of input RGB colors is maintained. www.sgi.com /misc/grafica/matrix   (921 words)

Basic Matrix Operations MATRIX : "A" row/col 1 2 3 4 units 1 3.78000e+00 9.70000e+00 -4.70000e+00 1.05000e+01 2 0.00000e+00 -5.80000e+00 2.00000e-01 -9.34000e+00 Max(A) = 10.5 Min(A) = -9.34 FUNCTION PURPOSE =============================================================== Min (A) Return a (1x1) matrix containing the minimum matrix element in matrix "A". MATRIX : "testVector" row/col 1 2 3 4 units 1 1.00000e+00 2.00000e+00 3.00000e+00 4.00000e+00 L2 norm of testVector is : 5.477 www.isr.umd.edu /~austin/aladdin.d/matrix-opers.html   (921 words)

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