section2_3
... If the number of rows equals the number of columns, then matrix X is a square matrix. If not, then matrix X is not a square matrix. ...
... If the number of rows equals the number of columns, then matrix X is a square matrix. If not, then matrix X is not a square matrix. ...
Fourier analysis on finite abelian groups
... of dimension n. First, a silly case: if all operators T ∈ H are scalar, then every vector is a simultaneous eigenvector for all the operators in H, and we are done. So now consider the (serious) case that not all operators in H are scalar. Let T ∈ H be a non-scalar operator. By the spectral theorem ...
... of dimension n. First, a silly case: if all operators T ∈ H are scalar, then every vector is a simultaneous eigenvector for all the operators in H, and we are done. So now consider the (serious) case that not all operators in H are scalar. Let T ∈ H be a non-scalar operator. By the spectral theorem ...
Isolated points, duality and residues
... following points: a complete characterization of the elements constructed at each integration, a better complexity bound (of the same order than Gaussian elimination in the vector space B on n2 + m matrices), bounds for the size of the coefficients of a basis of B̂, an explicit and effective corresp ...
... following points: a complete characterization of the elements constructed at each integration, a better complexity bound (of the same order than Gaussian elimination in the vector space B on n2 + m matrices), bounds for the size of the coefficients of a basis of B̂, an explicit and effective corresp ...
KNOT SIGNATURE FUNCTIONS ARE INDEPENDENT 1
... Proof of Theorem 1. For a given Seifert matrix V , σω (V ) can be viewed as an integer-valued function of ω ∈ S. Simple arguments show that jumps of this function can occur only at those values of ω that are roots of ∆V (t), and if the root is simple the jump is nontrivial. Also, for each V , σω (V ...
... Proof of Theorem 1. For a given Seifert matrix V , σω (V ) can be viewed as an integer-valued function of ω ∈ S. Simple arguments show that jumps of this function can occur only at those values of ω that are roots of ∆V (t), and if the root is simple the jump is nontrivial. Also, for each V , σω (V ...
General Linear Systems
... Gaussian Elimination with Complete Pivoting is STABLE. No significant reason to choose Complete pivoting over Partial pivoting. ...
... Gaussian Elimination with Complete Pivoting is STABLE. No significant reason to choose Complete pivoting over Partial pivoting. ...
Blocked Schur Algorithms for Computing the Matrix Square Root
... When used with full (non-triangular) matrices, more modest speedups are expected because of the significant overhead in computing the Schur decomposition. Figure 2 compares run times of the MATLAB function sqrtm (which does not use any blocking) and Fortran implementations of the the point method (f ...
... When used with full (non-triangular) matrices, more modest speedups are expected because of the significant overhead in computing the Schur decomposition. Figure 2 compares run times of the MATLAB function sqrtm (which does not use any blocking) and Fortran implementations of the the point method (f ...
C.6 Adjoints for Operators on a Hilbert Space
... Definition C.47. Let A ∈ B(H) be given. Then a closed subspace M ⊆ H is invariant under A if A(M ) ⊆ M, where A(M ) = {Ax : x ∈ M }. Note that it we do not require that A(M ) be equal to M . The simplest example of an invariant subspace is M = span{f }, where f is an eigenvector of A. Exercise C.48. ...
... Definition C.47. Let A ∈ B(H) be given. Then a closed subspace M ⊆ H is invariant under A if A(M ) ⊆ M, where A(M ) = {Ax : x ∈ M }. Note that it we do not require that A(M ) be equal to M . The simplest example of an invariant subspace is M = span{f }, where f is an eigenvector of A. Exercise C.48. ...
Minimal spanning and maximal independent sets, Basis
... Standardly, a spanning subset S 0 ⊆ S is called MINIMAL if every proper subset S 00 ⊂ S 0 does not span S. Lemma 6 Given a set of vectors S and a subset S 0 ⊆ S, the following three properties are equivalent: S 0 is independent and spans S, S 0 is a minimal spanning subsets of S, S 0 is a maximal in ...
... Standardly, a spanning subset S 0 ⊆ S is called MINIMAL if every proper subset S 00 ⊂ S 0 does not span S. Lemma 6 Given a set of vectors S and a subset S 0 ⊆ S, the following three properties are equivalent: S 0 is independent and spans S, S 0 is a minimal spanning subsets of S, S 0 is a maximal in ...
Google PageRank with stochastic matrix
... 2. If P is a n × n column-stochastic matrix, then kP k = 1. 3. If P is a column-stochastic matrix, then ρ(P ) = 1. 4. (Theorem:(Perron, 1907; Frobenius, 1912)): If P is a column-stochastic matrix and P be irreducible, in the sense that pij > 0 ∀ i, j ∈ S, then 1 is a simple eigenvalue of P . Moreove ...
... 2. If P is a n × n column-stochastic matrix, then kP k = 1. 3. If P is a column-stochastic matrix, then ρ(P ) = 1. 4. (Theorem:(Perron, 1907; Frobenius, 1912)): If P is a column-stochastic matrix and P be irreducible, in the sense that pij > 0 ∀ i, j ∈ S, then 1 is a simple eigenvalue of P . Moreove ...
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.