Miscellaneous matrix functions¶
- sage.matrix.matrix_misc.permanental_minor_polynomial(A, permanent_only=False, var='t', prec=None)[source]¶
Return the polynomial of the sums of permanental minors of
A.INPUT:
A– a matrixpermanent_only– ifTrue, return only the permanent of \(A\)var– name of the polynomial variableprec– if prec is not None, truncate the polynomial at precision \(prec\)
The polynomial of the sums of permanental minors is
\[\sum_{i=0}^{min(nrows, ncols)} p_i(A) x^i\]where \(p_i(A)\) is the \(i\)-th permanental minor of \(A\) (that can also be obtained through the method
permanental_minor()viaA.permanental_minor(i)).The algorithm implemented by that function has been developed by P. Butera and M. Pernici, see [BP2015]. Its complexity is \(O(2^n m^2 n)\) where \(m\) and \(n\) are the number of rows and columns of \(A\). Moreover, if \(A\) is a banded matrix with width \(w\), that is \(A_{ij}=0\) for \(|i - j| > w\) and \(w < n/2\), then the complexity of the algorithm is \(O(4^w (w+1) n^2)\).
INPUT:
A– matrixpermanent_only– boolean (default:False); ifTrue, only the permanent is computed (might be faster)var– a variable name
EXAMPLES:
sage: from sage.matrix.matrix_misc import permanental_minor_polynomial sage: m = matrix([[1,1],[1,2]]) sage: permanental_minor_polynomial(m) 3*t^2 + 5*t + 1 sage: permanental_minor_polynomial(m, permanent_only=True) 3 sage: permanental_minor_polynomial(m, prec=2) 5*t + 1
>>> from sage.all import * >>> from sage.matrix.matrix_misc import permanental_minor_polynomial >>> m = matrix([[Integer(1),Integer(1)],[Integer(1),Integer(2)]]) >>> permanental_minor_polynomial(m) 3*t^2 + 5*t + 1 >>> permanental_minor_polynomial(m, permanent_only=True) 3 >>> permanental_minor_polynomial(m, prec=Integer(2)) 5*t + 1
sage: M = MatrixSpace(ZZ,4,4) sage: A = M([1,0,1,0,1,0,1,0,1,0,10,10,1,0,1,1]) sage: permanental_minor_polynomial(A) 84*t^3 + 114*t^2 + 28*t + 1 sage: [A.permanental_minor(i) for i in range(5)] [1, 28, 114, 84, 0]
>>> from sage.all import * >>> M = MatrixSpace(ZZ,Integer(4),Integer(4)) >>> A = M([Integer(1),Integer(0),Integer(1),Integer(0),Integer(1),Integer(0),Integer(1),Integer(0),Integer(1),Integer(0),Integer(10),Integer(10),Integer(1),Integer(0),Integer(1),Integer(1)]) >>> permanental_minor_polynomial(A) 84*t^3 + 114*t^2 + 28*t + 1 >>> [A.permanental_minor(i) for i in range(Integer(5))] [1, 28, 114, 84, 0]
An example over \(\QQ\):
sage: M = MatrixSpace(QQ,2,2) sage: A = M([1/5,2/7,3/2,4/5]) sage: permanental_minor_polynomial(A, True) 103/175
>>> from sage.all import * >>> M = MatrixSpace(QQ,Integer(2),Integer(2)) >>> A = M([Integer(1)/Integer(5),Integer(2)/Integer(7),Integer(3)/Integer(2),Integer(4)/Integer(5)]) >>> permanental_minor_polynomial(A, True) 103/175
An example with polynomial coefficients:
sage: R.<a> = PolynomialRing(ZZ) sage: A = MatrixSpace(R,2)([[a,1], [a,a+1]]) sage: permanental_minor_polynomial(A, True) a^2 + 2*a
>>> from sage.all import * >>> R = PolynomialRing(ZZ, names=('a',)); (a,) = R._first_ngens(1) >>> A = MatrixSpace(R,Integer(2))([[a,Integer(1)], [a,a+Integer(1)]]) >>> permanental_minor_polynomial(A, True) a^2 + 2*a
A usage of the
varargument:sage: m = matrix(ZZ,4,[0,1,2,3,1,2,3,0,2,3,0,1,3,0,1,2]) sage: permanental_minor_polynomial(m, var='x') 164*x^4 + 384*x^3 + 172*x^2 + 24*x + 1
>>> from sage.all import * >>> m = matrix(ZZ,Integer(4),[Integer(0),Integer(1),Integer(2),Integer(3),Integer(1),Integer(2),Integer(3),Integer(0),Integer(2),Integer(3),Integer(0),Integer(1),Integer(3),Integer(0),Integer(1),Integer(2)]) >>> permanental_minor_polynomial(m, var='x') 164*x^4 + 384*x^3 + 172*x^2 + 24*x + 1
ALGORITHM:
The permanent \(perm(A)\) of a \(n \times n\) matrix \(A\) is the coefficient of the \(x_1 x_2 \ldots x_n\) monomial in
\[\prod_{i=1}^n \left( \sum_{j=1}^n A_{ij} x_j \right)\]Evaluating this product one can neglect \(x_i^2\), that is \(x_i\) can be considered to be nilpotent of order \(2\).
To formalize this procedure, consider the algebra \(R = K[\eta_1, \eta_2, \ldots, \eta_n]\) where the \(\eta_i\) are commuting, nilpotent of order \(2\) (i.e. \(\eta_i^2 = 0\)). Formally it is the quotient ring of the polynomial ring in \(\eta_1, \eta_2, \ldots, \eta_n\) quotiented by the ideal generated by the \(\eta_i^2\).
We will mostly consider the ring \(R[t]\) of polynomials over \(R\). We denote a generic element of \(R[t]\) by \(p(\eta_1, \ldots, \eta_n)\) or \(p(\eta_{i_1}, \ldots, \eta_{i_k})\) if we want to emphasize that some monomials in the \(\eta_i\) are missing.
Introduce an “integration” operation \(\langle p \rangle\) over \(R\) and \(R[t]\) consisting in the sum of the coefficients of the non-vanishing monomials in \(\eta_i\) (i.e. the result of setting all variables \(\eta_i\) to \(1\)). Let us emphasize that this is not a morphism of algebras as \(\langle \eta_1 \rangle^2 = 1\) while \(\langle \eta_1^2 \rangle = 0\)!
Let us consider an example of computation. Let \(p_1 = 1 + t \eta_1 + t \eta_2\) and \(p_2 = 1 + t \eta_1 + t \eta_3\). Then
\[p_1 p_2 = 1 + 2t \eta_1 + t (\eta_2 + \eta_3) + t^2 (\eta_1 \eta_2 + \eta_1 \eta_3 + \eta_2 \eta_3)\]and
\[\langle p_1 p_2 \rangle = 1 + 4t + 3t^2\]In this formalism, the permanent is just
\[perm(A) = \langle \prod_{i=1}^n \sum_{j=1}^n A_{ij} \eta_j \rangle\]A useful property of \(\langle . \rangle\) which makes this algorithm efficient for band matrices is the following: let \(p_1(\eta_1, \ldots, \eta_n)\) and \(p_2(\eta_j, \ldots, \eta_n)\) be polynomials in \(R[t]\) where \(j \ge 1\). Then one has
\[\langle p_1(\eta_1, \ldots, \eta_n) p_2 \rangle = \langle p_1(1, \ldots, 1, \eta_j, \ldots, \eta_n) p_2 \rangle\]where \(\eta_1,..,\eta_{j-1}\) are replaced by \(1\) in \(p_1\). Informally, we can “integrate” these variables before performing the product. More generally, if a monomial \(\eta_i\) is missing in one of the terms of a product of two terms, then it can be integrated in the other term.
Now let us consider an \(m \times n\) matrix with \(m \leq n\). The sum of permanental `k`-minors of `A` is
\[perm(A, k) = \sum_{r,c} perm(A_{r,c})\]where the sum is over the \(k\)-subsets \(r\) of rows and \(k\)-subsets \(c\) of columns and \(A_{r,c}\) is the submatrix obtained from \(A\) by keeping only the rows \(r\) and columns \(c\). Of course \(perm(A, \min(m,n)) = perm(A)\) and note that \(perm(A,1)\) is just the sum of all entries of the matrix.
The generating function of these sums of permanental minors is
\[g(t) = \left\langle \prod_{i=1}^m \left(1 + t \sum_{j=1}^n A_{ij} \eta_j\right) \right\rangle\]In fact the \(t^k\) coefficient of \(g(t)\) corresponds to choosing \(k\) rows of \(A\); \(\eta_i\) is associated to the \(i\)-th column; nilpotency avoids having twice the same column in a product of \(A\)’s.
For more details, see the article [BP2015].
From a technical point of view, the product in \(K[\eta_1, \ldots, \eta_n][t]\) is implemented as a subroutine in
prm_mul(). The indices of the rows and columns actually start at \(0\), so the variables are \(\eta_0, \ldots, \eta_{n-1}\). Polynomials are represented in dictionary form: to a variable \(\eta_i\) is associated the key \(2^i\) (or in Python1 << i). The keys associated to products are obtained by considering the development in base \(2\): to the monomial \(\eta_{i_1} \ldots \eta_{i_k}\) is associated the key \(2^{i_1} + \ldots + 2^{i_k}\). So the product \(\eta_1 \eta_2\) corresponds to the key \(6 = (110)_2\) while \(\eta_0 \eta_3\) has key \(9 = (1001)_2\). In particular all operations on monomials are implemented via bitwise operations on the keys.
- sage.matrix.matrix_misc.prm_mul(p1, p2, mask_free, prec)[source]¶
Return the product of
p1andp2, putting free variables inmask_freeto \(1\).This function is mainly use as a subroutine of
permanental_minor_polynomial().INPUT:
\(p1,p2\) – polynomials as dictionaries
mask_free– integer mask that give the list of free variables (the \(i\)-th variable is free if the \(i\)-th bit ofmask_freeis \(1\))prec– ifprecis notNone, truncate the product at precisionprec
EXAMPLES:
sage: from sage.matrix.matrix_misc import prm_mul sage: t = polygen(ZZ, 't') sage: p1 = {0: 1, 1: t, 4: t} sage: p2 = {0: 1, 1: t, 2: t} sage: prm_mul(p1, p2, 1, None) {0: 2*t + 1, 2: t^2 + t, 4: t^2 + t, 6: t^2}
>>> from sage.all import * >>> from sage.matrix.matrix_misc import prm_mul >>> t = polygen(ZZ, 't') >>> p1 = {Integer(0): Integer(1), Integer(1): t, Integer(4): t} >>> p2 = {Integer(0): Integer(1), Integer(1): t, Integer(2): t} >>> prm_mul(p1, p2, Integer(1), None) {0: 2*t + 1, 2: t^2 + t, 4: t^2 + t, 6: t^2}