Artifact 260ab4f4d4d00078022a4b6f43fbe660ccea45a07102300cdf7efe71e6fee603:
- File
r36/help/normform.tex
— part of check-in
[152fb3bdbb]
at
2011-10-17 17:58:33
on branch master
— svn:eol-style, svn:executable and line endings for files
in historical/r36 treegit-svn-id: https://svn.code.sf.net/p/reduce-algebra/code/trunk/historical@1480 2bfe0521-f11c-4a00-b80e-6202646ff360 (user: schoepf@users.sourceforge.net, size: 10215) [annotate] [blame] [check-ins using] [more...]
\section{Matrix Normal Forms} \begin{Operator}{Smithex} The operator \name{smithex} computes the Smith normal form S of a \nameref{matrix} A (say). It returns \{S,P,$P^-1$\} where $P*S*P^-1 = A$. \begin{Syntax} \name{smithex}\(\meta{matrix},\meta{variable}\) \meta{matrix} :- a rectangular \nameref{matrix} of univariate polynomials in \meta{variable}. \meta{variable} :- the variable. \end{Syntax} \begin{Examples} a := mat((x,x+1),(0,3*x^2)); & \begin{multilineoutput}{6cm} [x x + 1] [ ] a := [ 2 ] [0 3*x ] \end{multilineoutput}\\ smithex(a,x); & \begin{multilineoutput}{6cm} [1 0 ] [1 0] [x x + 1] \{ [ ], [ ], [ ] \} [ 3] [ 2 ] [ ] [0 x ] [3*x 1] [-3 -3 ] \end{multilineoutput}\\ \end{Examples} \end{Operator} \begin{Operator}{Smithex\_int} The operator \name{smithex\_int} performs the same task as \name{smithex} but on matrices containing only integer entries. Namely, \name{smithex\_int} returns \{S,P,$P^-1$\} where S is the smith normal form of the input \nameref{matrix} (A say), and $P*S*P^-1 = A$. \begin{Syntax} \name{smithex\_int}\(\meta{matrix}\) \meta{matrix} :- a rectangular \nameref{matrix} of integer entries. \end{Syntax} \begin{Examples} a := mat((9,-36,30),(-36,192,-180),(30,-180,180)); & \begin{multilineoutput}{6cm} [ 9 -36 30 ] [ ] a := [-36 192 -180] [ ] [30 -180 180 ] \end{multilineoutput}\\ smithex_int(a); & \begin{multilineoutput}{6cm} [3 0 0 ] [-17 -5 -4 ] [1 -24 30 ] [ ] [ ] [ ] \{ [0 12 0 ], [64 19 15 ], [-1 25 -30] \} [ ] [ ] [ ] [0 0 60] [-50 -15 -12] [0 -1 1 ] \end{multilineoutput}\\ \end{Examples} \end{Operator} \begin{Operator}{Frobenius} The operator \name{frobenius} computes the \name{frobenius} normal form F of a \nameref{matrix} (A say). It returns \{F,P,$P^-1$\} where $P*F*P^-1 = A$. \begin{Syntax} \name{frobenius}\(\meta{matrix}\) \meta{matrix} :- a square \nameref{matrix}. \end{Syntax} Field Extensions: By default, calculations are performed in the rational numbers. To extend this field the \nameref{arnum} package can be used. The package must first be loaded by load\_package arnum;. The field can now be extended by using the defpoly command. For example, defpoly sqrt2**2-2; will extend the field to include the square root of 2 (now defined by sqrt2). Modular Arithmetic: \name{Frobenius} can also be calculated in a modular base. To do this first type on modular;. Then setmod p; (where p is a prime) will set the modular base of calculation to p. By further typing on balanced\_mod the answer will appear using a symmetric modular representation. See \nameref{ratjordan} for an example. \begin{Examples} a := mat((x,x^2),(3,5*x)); & \begin{multilineoutput}{6cm} [ 2 ] [x x ] a := [ ] [3 5*x] \end{multilineoutput}\\ frobenius(a);& \begin{multilineoutput}{6cm} [ 2] [1 x] [ - x ] \{ [0 - 2*x ], [ ], [1 -----] \} [ ] [0 3] [ 3 ] [1 6*x ] [ ] [ 1 ] [0 --- ] [ 3 ] \end{multilineoutput}\\ load\_package arnum;\\ defpoly sqrt2**2-2;\\ a := mat((sqrt2,5),(7*sqrt2,sqrt2));& \begin{multilineoutput}{6cm} [ sqrt2 5 ] a := [ ] [7*sqrt2 sqrt2] \end{multilineoutput}\\ frobenius(a); & \begin{multilineoutput}{6cm} [0 35*sqrt2 - 2] [1 sqrt2 ] [ 1 ] \{ [ ], [ ], [1 - --- ] \} [1 2*sqrt2 ] [1 7*sqrt2] [ 7 ] [ ] [ 1 ] [0 ----*sqrt2] [ 14 ] \end{multilineoutput}\\ \end{Examples} \end{Operator} \begin{Operator}{Ratjordan} The operator \name{ratjordan} computes the rational Jordan normal form R of a \nameref{matrix} (A say). It returns \{R,P,$P^-1$\} where $P*R*P^-1 = A$. \begin{Syntax} \name{ratjordan}\(\meta{matrix}\) \meta{matrix} :- a square \nameref{matrix}. \end{Syntax} Field Extensions: By default, calculations are performed in the rational numbers. To extend this field the \name{arnum} package can be used. The package must first be loaded by load\_package arnum;. The field can now be extended by using the defpoly command. For example, defpoly sqrt2**2-2; will extend the field to include the square root of 2 (now defined by sqrt2). See \nameref{frobenius} for an example. Modular Arithmetic: \name{ratjordan} can also be calculated in a modular base. To do this first type on modular;. Then setmod p; (where p is a prime) will set the modular base of calculation to p. By further typing on balanced\_mod the answer will appear using a symmetric modular representation. \begin{Examples} a := mat((5,4*x),(2,x^2));& \begin{multilineoutput}{6cm} [5 4*x] [ ] a := [ 2 ] [2 x ] \end{multilineoutput}\\ ratjordan(a); & \begin{multilineoutput}{6cm} [0 x*( - 5*x + 8)] [1 5] [ -5 ] \{ [ ], [ ], [1 -----] \} [ 2 ] [0 2] [ 2 ] [1 x + 5 ] [ ] [ 1 ] [0 -----] [ 2 ] \end{multilineoutput}\\ on modular; \\ setmod 23; \\ a := mat((12,34),(56,78)); & \begin{multilineoutput}{6cm} [12 11] a := [ ] [10 9 ] \end{multilineoutput}\\ ratjordan(a); & \begin{multilineoutput}{6cm} [15 0] [16 8] [1 21] \{ [ ], [ ], [ ] \} [0 6] [19 4] [1 4 ] \end{multilineoutput}\\ on balanced\_mod;\\ ratjordan(a);& \begin{multilineoutput}{6cm} [- 8 0] [ - 7 8] [1 - 2] \{ [ ], [ ], [ ] \} [ 0 6] [ - 4 4] [1 4 ] \end{multilineoutput}\\ \end{Examples} \end{Operator} \begin{Operator}{Jordansymbolic} The operator \name{jordansymbolic} computes the Jordan normal form J of a \nameref{matrix} (A say). It returns \{J,L,P,$P^-1$\} where $P*J*P^-1 = A$. L = \{ll,mm\} where mm is a name and ll is a list of irreducible factors of p(mm). \begin{Syntax} \name{jordansymbolic}\(\meta{matrix}\) \meta{matrix} :- a square \nameref{matrix}. \end{Syntax} Field Extensions: By default, calculations are performed in the rational numbers. To extend this field the \nameref{arnum} package can be used. The package must first be loaded by load\_package arnum;. The field can now be extended by using the defpoly command. For example, defpoly sqrt2**2-2; will extend the field to include the square root of 2 (now defined by sqrt2). See \nameref{frobenius} for an example. Modular Arithmetic: \name{jordansymbolic} can also be calculated in a modular base. To do this first type on modular;. Then setmod p; (where p is a prime) will set the modular base of calculation to p. By further typing on balanced\_mod the answer will appear using a symmetric modular representation. See \nameref{ratjordan} for an example. % Extras: % If using \name{xr}, the X interface for REDUCE, then the appearance of the % output can be improved by switching on looking\_good. This % converts any lambdas to a greek font. \begin{Examples} a := mat((1,y),(2,5*y)); & \begin{multilineoutput}{6cm} [1 y ] a := [ ] [2 5*y] \end{multilineoutput}\\ jordansymbolic(a); & \begin{multilineoutput}{6cm} \{ [lambda11 0 ] [ ] [ 0 lambda12] , 2 {{lambda - 5*lambda*y - lambda + 3*y},lambda}, [lambda11 - 5*y lambda12 - 5*y] [ ] [ 2 2 ] , [ 2*lambda11 - 5*y - 1 5*lambda11*y - lambda11 - y + 1 ] [---------------------- ---------------------------------] [ 2 2 ] [ 25*y - 2*y + 1 2*(25*y - 2*y + 1) ] [ ] [ 2*lambda12 - 5*y - 1 5*lambda12*y - lambda12 - y + 1 ] [---------------------- ---------------------------------] [ 2 2 ] [ 25*y - 2*y + 1 2*(25*y - 2*y + 1) ] \} \end{multilineoutput}\\ \end{Examples} \end{Operator} \begin{Operator}{Jordan} The operator \name{jordan} computes the Jordan normal form J of a \nameref{matrix} (A say). It returns \{J,P,$P^-1$\} where $P*J*P^-1 = A$. \begin{Syntax} \name{jordan}\(\meta{matrix}\) \meta{matrix} :- a square \nameref{matrix}. \end{Syntax} Field Extensions: By default, calculations are performed in the rational numbers. To extend this field the \name{arnum} package can be used. The package must first be loaded by load\_package arnum;. The field can now be extended by using the defpoly command. For example, defpoly sqrt2**2-2; will extend the field to include the square root of 2 (now defined by sqrt2). See \nameref{frobenius} for an example. Modular Arithmetic: \name{Jordan} can also be calculated in a modular base. To do this first type on modular;. Then setmod p; (where p is a prime) will set the modular base of calculation to p. By further typing on balanced\_mod the answer will appear using a symmetric modular representation. See \nameref{ratjordan} for an example. \begin{Examples} a := mat((1,x),(0,x)); & \begin{multilineoutput}{6cm} [1 x] a := [ ] [0 x] \end{multilineoutput}\\ jordan(a);& \begin{multilineoutput}{6cm} \{ [1 0] [ ] [0 x] , [ 1 x ] [------- --------------] [ x - 1 2 ] [ x - 2*x + 1 ] [ ] [ 1 ] [ 0 ------- ] [ x - 1 ] , [x - 1 - x ] [ ] [ 0 x - 1] \} \end{multilineoutput}\\ \end{Examples} \end{Operator}