Problems 3.6 - Number Theory Web
... So AX = B is soluble for X if and only if B is a linear combination of the columns of A, that is B ∈ C(A). However by the first part of this question, B ∈ C(A) if and only if dim C([A|B]) = dim C(A), that is, rank [A|B] = rank A. 15. Let a1 , . . . , an be elements of F , not all zero. Let S denote ...
... So AX = B is soluble for X if and only if B is a linear combination of the columns of A, that is B ∈ C(A). However by the first part of this question, B ∈ C(A) if and only if dim C([A|B]) = dim C(A), that is, rank [A|B] = rank A. 15. Let a1 , . . . , an be elements of F , not all zero. Let S denote ...
Introduction to Matrices
... [4, 5, 6]T. Using row vectors from the beginning avoids all this weirdness. n More importantly, when we discuss how matrix multiplication can be used to perform coordinate space transformations, it will be convenient for the vector to be on the left and the matrix on the right. In this way, the tran ...
... [4, 5, 6]T. Using row vectors from the beginning avoids all this weirdness. n More importantly, when we discuss how matrix multiplication can be used to perform coordinate space transformations, it will be convenient for the vector to be on the left and the matrix on the right. In this way, the tran ...
Matrix Arithmetic
... 6. There is a unique m × n matrix Θ such that for any m × n matrix M, M + Θ = M. (This Θ is called the m × n zero matrix.) 7. For every m × n matrix M there is a unique m × n matrix N such that M + N = Θ. (This N is called the negative of M and is denoted −M.) Let’s prove something. How about that r ...
... 6. There is a unique m × n matrix Θ such that for any m × n matrix M, M + Θ = M. (This Θ is called the m × n zero matrix.) 7. For every m × n matrix M there is a unique m × n matrix N such that M + N = Θ. (This N is called the negative of M and is denoted −M.) Let’s prove something. How about that r ...
Matrix Factorization and Latent Semantic Indexing
... Problems with Lexical Semantics Ambiguity and association in natural language Polysemy: Words often have a multitude of meanings and different types of usage (more severe in very heterogeneous collections). The vector space model is unable to discriminate between different meanings of the same ...
... Problems with Lexical Semantics Ambiguity and association in natural language Polysemy: Words often have a multitude of meanings and different types of usage (more severe in very heterogeneous collections). The vector space model is unable to discriminate between different meanings of the same ...
a normal form in free fields - LaCIM
... 1. Introduction. Free fields, first introduced by Amitsur [A], were described by the first author as universal field of fractions of the ring of noncommutative polynomials; they are universal objects in the category whose morphisms are specializations. A characteristic property is that each full pol ...
... 1. Introduction. Free fields, first introduced by Amitsur [A], were described by the first author as universal field of fractions of the ring of noncommutative polynomials; they are universal objects in the category whose morphisms are specializations. A characteristic property is that each full pol ...
Sparse Matrices and Their Data Structures (PSC §4.2)
... I An accidental zero is a matrix element that is numerically zero but still occurs as a nonzero pair (i, 0) in the data structure. I Accidental zeros are created when a nonzero yi = −xi is added to a nonzero xi and the resulting zero is retained. I Testing all operations in a sparse matrix algorithm ...
... I An accidental zero is a matrix element that is numerically zero but still occurs as a nonzero pair (i, 0) in the data structure. I Accidental zeros are created when a nonzero yi = −xi is added to a nonzero xi and the resulting zero is retained. I Testing all operations in a sparse matrix algorithm ...
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.