Linear Algebra (wi1403lr)
... The invertible matrix theorem Let A be a square n × n matrix. Then the following statements are are either all true all false, that is, they are equivalent. a. A is an invertible matrix. j. There is an n × n matrix C such that CA = I . d. The equation Ax = 0 has only the trivial solution. c. A has ...
... The invertible matrix theorem Let A be a square n × n matrix. Then the following statements are are either all true all false, that is, they are equivalent. a. A is an invertible matrix. j. There is an n × n matrix C such that CA = I . d. The equation Ax = 0 has only the trivial solution. c. A has ...
On integer points in polyhedra: A lower bound
... comments see their paper [2]. For n = 2 and n = 3 there are examples in [6] and in [5] showing that PI can have as m a n y as f~(~on-l) vertices. Here we give such a construction for every n > 2. ...
... comments see their paper [2]. For n = 2 and n = 3 there are examples in [6] and in [5] showing that PI can have as m a n y as f~(~on-l) vertices. Here we give such a construction for every n > 2. ...
10 The Singular Value Decomposition
... that the corresponding left singular vector is flipped as well, and still obtain a valid SVD. Singular vectors must be flipped in pairs (a left vector and its corresponding right vector) because the singular values are required to be nonnegative. This is a trivial ambiguity. If desired, it can be re ...
... that the corresponding left singular vector is flipped as well, and still obtain a valid SVD. Singular vectors must be flipped in pairs (a left vector and its corresponding right vector) because the singular values are required to be nonnegative. This is a trivial ambiguity. If desired, it can be re ...
Analysis on arithmetic quotients Chapter I. The geometry of SL(2)
... algebraic tori—in effect, it conjugates the compact torus of SL2 (R) to the split torus, but inside SL2 (C). One application of the Cayley transform is to answer easily a basic question about non-Euclidean geometry: What is the non-Euclidean area of the non-Euclidean circle of radius r? The non-Eucl ...
... algebraic tori—in effect, it conjugates the compact torus of SL2 (R) to the split torus, but inside SL2 (C). One application of the Cayley transform is to answer easily a basic question about non-Euclidean geometry: What is the non-Euclidean area of the non-Euclidean circle of radius r? The non-Eucl ...
Matlab Tutorial I
... Matlab can work with groups of numbers arranged in rows and columns, called matricies. The simplest kind of matrix is made of a single row of numbers and is called a row matrix, or simply a vector. Matrices can be made up of real or complex numbers. You can create a matrix in several ways. One way i ...
... Matlab can work with groups of numbers arranged in rows and columns, called matricies. The simplest kind of matrix is made of a single row of numbers and is called a row matrix, or simply a vector. Matrices can be made up of real or complex numbers. You can create a matrix in several ways. One way i ...
Unit Overview - Connecticut Core Standards
... Once a familiarity with matrices and operations with matrices is established, the next investigations gradually build on matrix algebra, the notion of the determinant, the inverse of a matrix and procedures for finding the inverse. This allows for solving matrix equations and solving systems of line ...
... Once a familiarity with matrices and operations with matrices is established, the next investigations gradually build on matrix algebra, the notion of the determinant, the inverse of a matrix and procedures for finding the inverse. This allows for solving matrix equations and solving systems of line ...
Appendix E An Introduction to Matrix Algebra
... is known as a matrix. The numbers aij are called elements of the matrix, with the subscript i denoting the row and the subscript j denoting the column. A matrix with m rows and n columns is said to be a matrix of order (m, n) or alternatively called an m n (m-by-n) matrix.When the number of the co ...
... is known as a matrix. The numbers aij are called elements of the matrix, with the subscript i denoting the row and the subscript j denoting the column. A matrix with m rows and n columns is said to be a matrix of order (m, n) or alternatively called an m n (m-by-n) matrix.When the number of the co ...
![[2013 question paper]](http://s1.studyres.com/store/data/008844912_1-7494f7d6a29d900492173910cd64b54c-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.