Matrix

Matrix -- the class of all matrices for which Groebner basis operations are available from the engine.

A matrix is a map from a graded module to a graded module, see Module. The degree of the map is not necessarily 0, and may be obtained with degree.

Multiplication of matrices corresponds to composition of maps, and when f and g are maps so that the target Q of g equals the source P of f, the product f*g is defined, its source is the source of g, and its target is the target of f. The degree of f*g is the sum of the degrees of f and of g. The product is also defined when P != Q, provided only that P and Q are free modules of the same rank. If the degrees of P differ from the corresponding degrees of Q by the same degree d, then the degree of f*g is adjusted by d so it will have a good chance to be homogeneous, and the target and source of f*g are as before.

If h is a matrix then h_j is the j-th column of the matrix, and h_j_i is the entry in row i, column j. The notation h_(i,j) can be used as an abbreviation for h_j_i, allowing row and column indices to be written in the customary order.

If m and n are matrices, a is a ring element, and i is an integer, then 'm+n', 'm-n', '-m', 'm n', 'a m', and 'i m' denote the usual matrix arithmetic. 'm == n', and 'm == 0' are used to check equality of matrices.

Operations which produce matrices:

genericMatrix(R,x,r,c) -- an r by c generic matrix

genericSkewMatrix(R,x,r) -- an r by r generic skew matrix

genericSymmetricMatrix(R,x,r) -- an r by r generic symmetric matrix

id_F -- identity map F <--- F

matrix -- create a matrix

map -- create a map of modules

random(F,G) -- a random graded matrix F <-- G

Operations on matrices:

==

!=

+

-

*

^

%

//

f_(i,j) -- getting an entry

f_{i,j} -- extracting or permuting columns

f|g -- horizontal concatenation

|| -- vertical concatenation

m ++ n -- direct sum

Matrix ** Matrix -- tensor product of matrices

Matrix ** Module -- tensor product, degree shifting

:

substitute -- evaluation of a matrix of polynomials

borel m -- smallest Borel submodule containing lead monomials of m

compress -- removal of zero columns

contract(m,n) -- contraction of n by m (i.e. diff without the coefficients)

det -- determinant

diff(m,n) -- differentiation of n by m

divideByVariable -- divide columns by a variable repeatedly

selectInSubring

entries m -- the entries of m

exteriorPower(i,m) -- exterior power of m

flatten m -- the one row matrix with the entries of m

isHomogeneous m -- whether the matrix m is graded

isInjective m -- whether a map is injective

isIsomorphism m -- whether the map m is an isomorphism

isSurjective m -- whether a map is surjective

isWellDefined m -- whether a map is well-defined

homogenize m -- homogenize the marix m

jacobian m -- Jacobian matrix of m

koszul(i,m) -- i-th Koszul matrix of m

basis(deg,m) -- k-basis of a module in a given degree

leadTermMatrix m -- the lead monomial matrix of the columns of m

leadTermMatrix(i,m) -- the lead terms w.r.t the first i weight vectors of m

minors(i,m) -- ideal of i by i minors of m

pfaffians(i,m) -- ideal of i by i Pfaffians of the skew symmetric matrix m

ring m -- the base ring of the matrix m

source m -- the source freemodule (i.e., the columnspace) of m

submatrix(m,rows,cols) -- extract a submatrix

symmetricPower(i,m) -- i-th symmetric power of m

target m -- the target free module (i.e., the rowspace) of m

trace -- trace

transpose m -- transpose a matrix

Operations which produce modules or chain complexes from matrices:

cokernel m -- the cokernel of the matrix m

kernel m -- the kernel of the matrix m

Operations which produce Groebner bases from matrices:

gb

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