Geometric Means - College of William and Mary
... On taking (k + 1)th roots we have the bound in (3.6). Finally, consider properties P1–P9. These properties can be easily proved by induction and the fact that they are known to be true for k = 2. To illustrate that we prove P3: For k ≥ 2 we have G(A1 , . . . , Ak ) = G(Ai1 , . . . , Aik ) for any pe ...
... On taking (k + 1)th roots we have the bound in (3.6). Finally, consider properties P1–P9. These properties can be easily proved by induction and the fact that they are known to be true for k = 2. To illustrate that we prove P3: For k ≥ 2 we have G(A1 , . . . , Ak ) = G(Ai1 , . . . , Aik ) for any pe ...
Linear Algebra - UC Davis Mathematics
... of functions. In linear algebra, functions will again be the focus of your attention, but functions of a very special type. In precalculus you were perhaps encouraged to think of a function as a machine f into which one may feed a real number. For each input x this machine outputs a single real numb ...
... of functions. In linear algebra, functions will again be the focus of your attention, but functions of a very special type. In precalculus you were perhaps encouraged to think of a function as a machine f into which one may feed a real number. For each input x this machine outputs a single real numb ...
Algorithms for Matrix Canonical Forms
... The canonical Howell form was originally described by Howell (1986) for matrices over Z/(N ) but generalizes readily to matrices over an arbitrary principal ideal ring R. Howell’s proof of existence is constructive and leads to an O(n3 ) basic operations algorithm. When R is a field, the Howell form ...
... The canonical Howell form was originally described by Howell (1986) for matrices over Z/(N ) but generalizes readily to matrices over an arbitrary principal ideal ring R. Howell’s proof of existence is constructive and leads to an O(n3 ) basic operations algorithm. When R is a field, the Howell form ...
Brauer-Thrall for totally reflexive modules
... (1.2) Theorem. If there exists a totally reflexive R-module without free summands, which is presented by a matrix that has a column or a row with only one non-zero entry, then that entry is an exact zero divisor in R. These two results—the latter of which is distilled from Theorem (5.3)—show that ex ...
... (1.2) Theorem. If there exists a totally reflexive R-module without free summands, which is presented by a matrix that has a column or a row with only one non-zero entry, then that entry is an exact zero divisor in R. These two results—the latter of which is distilled from Theorem (5.3)—show that ex ...
On a different kind of d -orthogonal polynomials that generalize the Laguerre polynomials
... The d-orthogonality notion seems to appear in various domains of mathematics. For instance, there is a closed relationship between 2-orthogonality and the birth and the death process [26]. Furthermore, Vinet and Zhedanov [24] showed that there exists a connection with application of d-orthogonal pol ...
... The d-orthogonality notion seems to appear in various domains of mathematics. For instance, there is a closed relationship between 2-orthogonality and the birth and the death process [26]. Furthermore, Vinet and Zhedanov [24] showed that there exists a connection with application of d-orthogonal pol ...
Counting Regions in Hyperplane Arrangements
... It is not hard to see that this conjecture holds for t = 1 and n − 1. It has also been shown for the second partial sum, and for special classes of graphs including trees, split graphs, regular graphs, co-graphs, and all graphs with at most 10 vertices.1 Brouwer’s conjecture is intimately connected ...
... It is not hard to see that this conjecture holds for t = 1 and n − 1. It has also been shown for the second partial sum, and for special classes of graphs including trees, split graphs, regular graphs, co-graphs, and all graphs with at most 10 vertices.1 Brouwer’s conjecture is intimately connected ...
Coding theory lectures
... x whose weight is even but not divisible by four. Furthermore C1 consists of all codewords of C whose weights are not divisible by four. Proof. Since C is self-orthogonal then all codewords have even weight, then either (i) holds or there exists a codeword x of even weight but not of weight a multip ...
... x whose weight is even but not divisible by four. Furthermore C1 consists of all codewords of C whose weights are not divisible by four. Proof. Since C is self-orthogonal then all codewords have even weight, then either (i) holds or there exists a codeword x of even weight but not of weight a multip ...
Lie Algebras
... Lie algebras are vector spaces endowed with a special non-associative multiplication called a Lie bracket. They arise naturally in the study of mathematical objects called Lie groups, which serve as groups of transformations on spaces with certain symmetries. An example of a Lie group is the group O ...
... Lie algebras are vector spaces endowed with a special non-associative multiplication called a Lie bracket. They arise naturally in the study of mathematical objects called Lie groups, which serve as groups of transformations on spaces with certain symmetries. An example of a Lie group is the group O ...
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.