Chapter 4. Drawing lines: conditionals and coordinates in PostScript
... if we want to draw the visible part of the line x + y = 1. One reason this is not quite a trivial problem is that we are certainly not able to draw the entire infinite line. There is essentially only one way to draw parts of a line in PostScript, and that is to use @moveto@ and @lineto@ to draw a se ...
... if we want to draw the visible part of the line x + y = 1. One reason this is not quite a trivial problem is that we are certainly not able to draw the entire infinite line. There is essentially only one way to draw parts of a line in PostScript, and that is to use @moveto@ and @lineto@ to draw a se ...
9. Numerical linear algebra background
... Solving linear equations by LU factorization. given a set of linear equations Ax = b, with A nonsingular. 1. LU factorization. Factor A as A = P LU ((2/3)n3 flops). 2. Permutation. Solve P z1 = b (0 flops). 3. Forward substitution. Solve Lz2 = z1 (n2 flops). 4. Backward substitution. Solve U x = z2 ...
... Solving linear equations by LU factorization. given a set of linear equations Ax = b, with A nonsingular. 1. LU factorization. Factor A as A = P LU ((2/3)n3 flops). 2. Permutation. Solve P z1 = b (0 flops). 3. Forward substitution. Solve Lz2 = z1 (n2 flops). 4. Backward substitution. Solve U x = z2 ...
Free Probability Theory and Random Matrices
... However: we might expect that we have almost sure convergence to a deterministic result • if N → ∞ and • if the eigenspaces are almost surely in a ”typical” or ”generic” position. ...
... However: we might expect that we have almost sure convergence to a deterministic result • if N → ∞ and • if the eigenspaces are almost surely in a ”typical” or ”generic” position. ...
Structured ring spectra and displays
... Fej = V −1 (v(1) · ej ) = (pbij )ei for (d + 1) ≤ j ≤ h, ...
... Fej = V −1 (v(1) · ej ) = (pbij )ei for (d + 1) ≤ j ≤ h, ...
AIMS Lecture Notes 2006 4. Gaussian Elimination Peter J. Olver
... the algorithm successfully reduces it to upper triangular form U with all non-zero pivots on the diagonal. In other words, for regular matrices, as the algorithm proceeds, each successive pivot appearing on the diagonal must be nonzero; otherwise, the matrix is not regular. We then use the pivot row ...
... the algorithm successfully reduces it to upper triangular form U with all non-zero pivots on the diagonal. In other words, for regular matrices, as the algorithm proceeds, each successive pivot appearing on the diagonal must be nonzero; otherwise, the matrix is not regular. We then use the pivot row ...
Las Vegas algorithms for matrix groups
... Proof: G acts projectively on a vector space of dimension d over a q element field, where qd is polynomially bounded by u(G). Also, G contains a subgroup H isomorphic to G1 x G2, where G I is the same kind of classical group as G acting projectively on a vector space of dimension d - 4, and G2 is no ...
... Proof: G acts projectively on a vector space of dimension d over a q element field, where qd is polynomially bounded by u(G). Also, G contains a subgroup H isomorphic to G1 x G2, where G I is the same kind of classical group as G acting projectively on a vector space of dimension d - 4, and G2 is no ...
Notes
... We’ve already observed that O (n) ⊆ Rn , and that it is a compact submanifold. Quick Exercise 1.9. What is O (1)? Incidentally, as you’ve just observed in the n = 1 case, O (n) is not a connected manifold – it splits into two pieces: SO (n) and what’s sometimes called SO− (n), the set of matrices U ...
... We’ve already observed that O (n) ⊆ Rn , and that it is a compact submanifold. Quick Exercise 1.9. What is O (1)? Incidentally, as you’ve just observed in the n = 1 case, O (n) is not a connected manifold – it splits into two pieces: SO (n) and what’s sometimes called SO− (n), the set of matrices U ...
October 17, 2011 THE ELGAMAL CRYPTOSYSTEM OVER
... same size, with half the computational cost. In this paper, we denote the group of non-singular circulant matrices of size d by C(d, q) and the group of special circulant matrices, i.e., circulant matrices with determinant 1, by SC(d, q) respectively. Let us pause here and discuss, what is a better ...
... same size, with half the computational cost. In this paper, we denote the group of non-singular circulant matrices of size d by C(d, q) and the group of special circulant matrices, i.e., circulant matrices with determinant 1, by SC(d, q) respectively. Let us pause here and discuss, what is a better ...
On the Kemeny constant and stationary distribution vector
... distribution vector when A is perturbed, with small values of the Kemeny constant corresponding to well–conditioned stationary distribution vectors. In view of these last observations regarding low values of the Kemeny constant, it is not surprising that there is interest in identifying stochastic m ...
... distribution vector when A is perturbed, with small values of the Kemeny constant corresponding to well–conditioned stationary distribution vectors. In view of these last observations regarding low values of the Kemeny constant, it is not surprising that there is interest in identifying stochastic m ...
Honors Algebra 4, MATH 371 Winter 2010
... M to be Tor(M ) := {m ∈ M : annR (m) 6= 0}. We say that an R-module N is torsion free if Tor(N ) = 0. Prove that Tor(M ) really is a submodule of M and that the quotient M/ Tor(M ) is torsion free. Solution: Suppose m, n ∈ Tor(M ) and that r, s ∈ R \ {0} are elements of annR (m) and annR (n), respec ...
... M to be Tor(M ) := {m ∈ M : annR (m) 6= 0}. We say that an R-module N is torsion free if Tor(N ) = 0. Prove that Tor(M ) really is a submodule of M and that the quotient M/ Tor(M ) is torsion free. Solution: Suppose m, n ∈ Tor(M ) and that r, s ∈ R \ {0} are elements of annR (m) and annR (n), respec ...
![arXiv:math/0612264v3 [math.NA] 28 Aug 2007](http://s1.studyres.com/store/data/017761287_1-ffc05f8355097f4d61a5ecaa6e5ed82e-300x300.png)
Jordan normal form
In linear algebra, a Jordan normal form (often called Jordan canonical form)of a linear operator on a finite-dimensional vector space is an upper triangular matrix of a particular form called a Jordan matrix, representing the operator with respect to some basis. Such matrix has each non-zero off-diagonal entry equal to 1, immediately above the main diagonal (on the superdiagonal), and with identical diagonal entries to the left and below them. If the vector space is over a field K, then a basis with respect to which the matrix has the required form exists if and only if all eigenvalues of the matrix lie in K, or equivalently if the characteristic polynomial of the operator splits into linear factors over K. This condition is always satisfied if K is the field of complex numbers. The diagonal entries of the normal form are the eigenvalues of the operator, with the number of times each one occurs being given by its algebraic multiplicity.If the operator is originally given by a square matrix M, then its Jordan normal form is also called the Jordan normal form of M. Any square matrix has a Jordan normal form if the field of coefficients is extended to one containing all the eigenvalues of the matrix. In spite of its name, the normal form for a given M is not entirely unique, as it is a block diagonal matrix formed of Jordan blocks, the order of which is not fixed; it is conventional to group blocks for the same eigenvalue together, but no ordering is imposed among the eigenvalues, nor among the blocks for a given eigenvalue, although the latter could for instance be ordered by weakly decreasing size.The Jordan–Chevalley decomposition is particularly simple with respect to a basis for which the operator takes its Jordan normal form. The diagonal form for diagonalizable matrices, for instance normal matrices, is a special case of the Jordan normal form.The Jordan normal form is named after Camille Jordan.