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This package contains routines for computing the following normal forms of matrices:
When are two given matrices similar? Similar matrices have the same trace, determinant, characteristic polynomial, and eigenvalues, but the matrices
are the same in all four of the above but are not similar. Otherwise there could exist a
nonsingular ∈M2 (the set of all 2 × 2 matrices) such that
=
−1 =
0
−1 = 0,
which is a contradiction since
≠0.
Two matrices can look very different but still be similar. One approach to determining whether two given matrices are similar is to compute the normal form of them. If both matrices reduce to the same normal form they must be similar.
NORMFORM is a package for computing the following normal forms of matrices:
The package is loaded by load_package normform;
By default all calculations are carried out in Q (the rational numbers). For smithex, frobenius, ratjordan, jordansymbolic, and jordan, this field can be extended. Details are given in the respective sections.
The frobenius, ratjordan, and jordansymbolic normal forms can also be computed in a modular base. Again, details are given in the respective sections.
The algorithms for each routine are contained in the source code.
NORMFORM has been converted from the normform and Normform packages written by T.M.L. Mulders and A.H.M. Levelt. These have been implemented in Maple [4].
smithex(,x) computes the Smith normal form
of the matrix
.
It returns {,
,
−1} where
,
, and
−1 are such that
−1 =
.
is a rectangular matrix of univariate polynomials in x.
x is the variable name.
Calculations are performed in . To extend this field the ARNUM package can be used.
For details see section 8.
Hence, if we have the case that n = m, as well as rank() = n, then product
(
(i,i),i = 1…n) = det(
) / lcoeff(det(
),x).
load_package normform;
Given an n by m rectangular matrix that contains only integer entries,
smithex_int(
) computes the Smith normal form
of
.
It returns {,
,
−1} where
,
, and
−1 are such that
−1 =
.
Hence, if we have the case that n = m, as well as rank() = n, then abs(det(
)) =
product(
(i,i),i = 1…n).
load_package normform;
smithex_int() =
frobenius() computes the Frobenius normal form
of the matrix
.
It returns {,
,
−1} where
,
, and
−1 are such that
−1 =
.
is a square matrix.
Calculations are performed in . To extend this field the ARNUM package can be used.
For details see section 8.
frobenius can be calculated in a modular base. For details see section 9.
where the (pi)’s are companion matrices associated with polynomials
p1,p2,…,pk, with the property that pi divides pi+1 for i = 1…k − 1. All unmarked
entries are zero.
load_package normform;
frobenius() =
ratjordan() computes the rational Jordan normal form
of the matrix
.
It returns {,
,
−1} where
,
, and
−1 are such that
−1 =
.
is a square matrix.
Calculations are performed in . To extend this field the ARNUM package can be used.
For details see section 8.
ratjordan can be calculated in a modular base. For details see section 9.
The rij’s have the following shape:
where there are eij times (p) blocks along the diagonal and
(p) is the
companion matrix associated with the irreducible polynomial p. All unmarked
entries are zero.
load_package normform;
ratjordan() =
jordansymbolic() computes the Jordan normal form
of the matrix
.
It returns {,
,
,
−1}, where
,
, and
−1 are such that
−1 =
.
= { ll , ξ
}, where ξ is a name and ll is a list of irreducible factors of p(ξ).
is a square matrix.
Calculations are performed in . To extend this field the ARNUM package can be used.
For details see section 8.
jordansymbolic can be calculated in a modular base. For details see subsection 9.
If using xr, the X interface for REDUCE, the appearance of the output can be improved by switching on looking_good;. This converts all lambda to ξ and improves the indexing, eg: lambda12 ⇒ ξ12. The example (section 6.6) shows the output when this switch is on.
There are k − 1 terms “+1” in the superdiagonal; the scalar λ appears k times on the main diagonal. All other matrix entries are zero, and ȷ1(λ) = (λ).
in which the orders ni may not be distinct and the values λi need not be distinct.
load_package normform;
on looking_good;
jordansymbolic() =
solve(−y3 + xi2 − 4 ∗ xi + 3,xi);
{ξ = + 2,ξ = −
+ 2}
= sub({xi(1,1) = sqrt(y3 + 1) + 2,xi(1,2) = −sqrt(y3 + 1) + 2},
first jordansymbolic ());
For a similar example ot this in standard REDUCE (ie: not using xr), see the normform.log file.
jordan() computes the Jordan normal form
of the matrix
.
It returns {,
,
−1}, where
,
, and
−1 are such that
−1 =
.
is a square matrix.
Calculations are performed in . To extend this field the ARNUM package can be used.
For details see section 8.
In certain polynomial cases fullroots is turned on to compute the zeroes. This can lead to the calculation taking a long time, as well as the output being very large. In this case a message ***** WARNING: fullroots turned on. May take a while. will be printed. It may be better to kill the calculation and compute jordansymbolic instead.
There are k − 1 terms “+1” in the superdiagonal; the scalar λ appears k times on the main diagonal. All other matrix entries are zero, and ȷ1(λ) = (λ).
in which the orders ni may not be distinct and the values λi need not be distinct.
load_package normform;
= first jordan(
);
The package is loaded by load_package arnum;. The algebraic field can now be
extended. For example, defpoly sqrt2**2-2; will extend it to include
(defined here by sqrt2). The ARNUM package was written by Eberhard Schrüfer and is
described in the arnum.tex file.
load_package normform;
load_package arnum;
defpoly sqrt2**2-2;
(sqrt2 now changed to for looks!)
Calculations can be performed in a modular base by switching on modular;. The
base can then be set by setmod p; (p a prime). The normal form will then have entries
in ∕p
.
By also switching on balanced_mod; the output will be shown using a symmetric modular representation.
Information on this modular manipulation can be found in chapter 9 (Polynomials and Rationals) of the REDUCE User’s Manual [5].
load_package normform;
on modular;
setmod 23;
jordansymbolic() =
on balanced_mod;
jordansymbolic() =
[1] T.M.L.Mulders and A.H.M. Levelt: The Maple normform and Normform packages. (1993)
[2] T.M.L.Mulders: Algoritmen in De Algebra, A Seminar on Algebraic Algorithms, Nigmegen. (1993)
[3] Roger A. Horn and Charles A. Johnson: Matrix Analysis. Cambridge University Press (1990)
[4] Bruce W. Chat…[et al.]: Maple (Computer Program). Springer-Verlag (1991)
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