![Table of Contents 1 Introduction to Vectors](http://s1.studyres.com/store/data/008688459_1-bd46d8f26d2519ce3e40cea11781d51e-300x300.png)
The columns of AB are combinations of the columns of A. The
... are linearly independent because the corresponding columns of B are linearly independent, and because row operations do not change linear dependencies between the columns. Now we know that the column space of A is spanned by the 5 columns of A. But we can see in B that the third column is a linear c ...
... are linearly independent because the corresponding columns of B are linearly independent, and because row operations do not change linear dependencies between the columns. Now we know that the column space of A is spanned by the 5 columns of A. But we can see in B that the third column is a linear c ...
matrix - O6U E-learning Forum
... • Any matrix in which every element is zero is called a zero or null matrix. If A is a zero matrix, we can simply write A= 0 A + 0 = 0 + A = A A–A= 0 ...
... • Any matrix in which every element is zero is called a zero or null matrix. If A is a zero matrix, we can simply write A= 0 A + 0 = 0 + A = A A–A= 0 ...
TOPOLOGICALLY UNREALIZABLE AUTOMORPHISMS OF FREE
... of finite index with <¡>(S)= S. I suspect that <¡>\Sis not realizable by a surface homeomorphism and that this can be proved by examining eigenvalues. If S is a normal subgroup andinduces the identity on G = F/S, then
... of finite index with <¡>(S)= S. I suspect that <¡>\Sis not realizable by a surface homeomorphism and that this can be proved by examining eigenvalues. If S is a normal subgroup and
A Tutorial on MATLAB Objective: To generate arrays in MATLAB
... 6.2 Take the transpose of matrices below; a. b. c. d. 7. Objective: Array operations such as element-by-element multiplications, and so on. Procedure: Dot (.) operator before an operator such as ', ^, * represents the array operations for the matrix operations. The list of array operators is shown b ...
... 6.2 Take the transpose of matrices below; a. b. c. d. 7. Objective: Array operations such as element-by-element multiplications, and so on. Procedure: Dot (.) operator before an operator such as ', ^, * represents the array operations for the matrix operations. The list of array operators is shown b ...
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.