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Computing The Smith Forms Of Integer Matrices And - Bad Request
Get Computing The Smith Forms Of Integer Matrices And - Bad Request
Ents for the degree of Doctor of Philosophy in Computer Science Summer 2005 c 2005 Zhendong Wan All Rights Reserved COMPUTING THE SMITH FORMS OF INTEGER MATRICES AND SOLVING RELATED PROBLEMS by Zhendong Wan Approved: Henry R. Glyde, Ph.D. Interim Chair of the Department of Computer and Information Sciences Approved: Thomas M. Apple, Ph.D. Dean of the College of Arts and Sciences Approved: Conrado M. Gempesaw II, Ph.D. Vice Provost for Academic and International Programs I certify that.
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Efcient FAQ
Normal form of a matrix is a matrix satisfying following conditions: consist of only ones and zeros. every row has a maximum of single one and rest are all zeros (there can be rows with all zeros). Rank of a matrix can be found from its normal form by counting the number of rows with non zero elements.
The determinant of a 3×3 matrix with integer entries is an integer. that all linear combinations of elements in Z yield another element of Z. Thus, since the determinant of a matrix with integer values is a linear combination of integers, it must also be an integer.
If A is a matrix with integer coefficients, we can transform A into Smith normal form using row and column operations. A = [ d 0 0 B ] , where B is a k − 1 by n − 1 matrix with integer entries, all divisible by d, and the 2 0′s denote zero vectors of the appropriate size..
We give a proof by induction that any complex-valued square matrix A may be put in Jordan normal form. Since the underlying vector space can be shown to be the direct sum of invariant subspaces associated with the eigenvalues, A can be assumed to have just one eigenvalue λ.
The Smith normal form of a matrix is diagonal, and can be obtained from the original matrix by multiplying on the left and right by invertible square matrices. In particular, the integers are a PID, so one can always calculate the Smith normal form of an integer matrix.
Hint: The normal form of a matrix is obtained from its original matrix by undergoing transformations on the rows and columns. The transformations include multiplying a row with a certain integer and subtracting the values of the row from another row and placing the result in its previous place.
The Smith form S is a diagonal matrix. [ U , V , S ] = smithForm( A ) returns the Smith normal form of A and unimodular transformation matrices U and V , such that S = U*A*V . ___ = smithForm( A , var ) assumes that the elements of A are univariate polynomials in the specified variable var .
The main result about Smith normal form, of course, is that every integer matrix has one. It is unique up to signs. There is an algorithm to compute it which is cross between row reduction and the Euclidean algorithm for computing greatest common divisors.
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