permutation matrix P has the rows of the identity I in any order. $$P_,$ would swap the columns $1$ and $3. permutation matrix is a square matrix that rearranges the rows of a n other matrix by multiplication. If we want to exchange rows $n$ and $m,$ we need to swap the corresponding rows of the $I$ matrix: If $m=3$ and $n=1,$ $IA,$ with $I$ being the identity matrix, selects every row of $A$ and leaves it in its place. Pv ≔ CreatePermutation nv, output = ' Vector 'Ī ≔ A convert pv, list, 1. Pm ≔ CreatePermutation nv, output = ' Matrix ' Nv, U ≔ LUDecomposition A, output = ' NAG ' However, it can always be accessed through the long form of the command by using LinearAlgebra(.). This function is part of the LinearAlgebra package, and so it can be used in the form CreatePermutation(.) only after executing the command with(LinearAlgebra). If a constructor option is provided in both the calling sequence directly and in an outputoptions option, the latter takes precedence (regardless of the order). These options may also be provided in the form outputoptions=, where represents a Maple list. The determinant of a permutation matrix is either or 1 and equals Signature permv. Permutation matrices are closed under matrix multiplication, so is again a permutation matrix. The constructor options provide additional information (readonly, shape, storage, order, datatype, and attributes) to the Matrix or Vector constructor that builds the result. A permutation matrix is an orthogonal matrix, where the inverse is equivalent to the transpose. The simplest permutation matrix is I, the identity matrix. For example, the pivot vector Vector(3, ) is equivalent to the pivot vector Vector(6, ). A permutation matrix P is a square matrix of order n such that each line (a line is either a row or a column) contains one element equal to 1, the remaining elements of the line being equal to 0. This allows the shortened form of a pivot vector used by NAG when the number of rows of an LU factorizable Matrix A is larger than the number of its columns. This value must be greater than or equal to the dimension of the input Vector V. If the optional nonnegative integer parameter d is provided then the dimension(s) of the output object have the value d. The default datatype of a returned permutation Matrix is integer. A (i.e., no pivoting is required to compute the decomposition of M. Premultiplying A by M permutes A to a Matrix whose factorization would require no row swapping in order to use the same choices of pivot values. Ī returned permutation Matrix M has all entries with value 1 or 0. The default datatype of a returned permutation Vector is integer. n, then no swapping would be required to do the factorization with the exact same choices of pivot rows. Prior to performing the i th pivot, row i and row V of the partially row-reduced Matrix are swapped.Ī returned permutation Vector U has as its i th entry the ordinal of the row of A such that, if all U th rows of A were permuted to the i th row, i = 1. Its i th element is the ordinal of the row of the partially row-reduced Matrix which is selected as the i th choice of pivoting row. For example, the identity permutation (1,2.,n) is even (it is obtained. Thus a permutation is called evenif an even number of transpositions is required, and oddotherwise. By default, the resulting object is a Vector of rectangular storage and integer datatype or a Matrix of sparse storage and integer datatype.Ī pivot vector V in NAG form, returned for example by an LU decomposition of a Matrix A, has all integer entries. The number of required transpositions to obtain a given permutation may depend on the way we do it, but the parityof this number depends only on this given permutation. The CreatePermutation(V) function constructs a permutation Vector or Matrix from a NAG pivot vector. (optional) constructor options for the result object (optional) equation of the form output = obj where obj is one of 'Vector' or 'Matrix', or a list containing one of these names selects format of the output object (optional) nonnegative integer dimension(s) of output Convert a NAG pivot vector to a permutation Vector or Matrix
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