One Hat Cyber Team

Edit File: SuperLinearAlgebra.m2

newPackage(
    "SuperLinearAlgebra", 
    Version => "0.1", 
    Date => "29 January 2021", 
    Authors => {
        {Name => "Fereshteh Bahadorykhalily", 
         Email => "f.bahadori.khalili@gmail.com", 
         HomePage => "https://www.researchgate.net/profile/Fereshte_Bahadorykhalily"
        }, 
        {Name => "Fatemeh Tarashi Kashani", 
         Email => "tarashikashanifatemeh@gmail.com", 
         HomePage => "https://www.linkedin.com/in/fatemehtarashi/"
        }
    }, 
    Headline => "computations related to supermatrices", 
    DebuggingMode => false,
    Keywords => {"Linear Algebra"}
)

--------------------
-- Exports
--------------------

export {
   -- Types
    "SuperMatrix", 
    
   -- Methods
    "berezinian", 
    "inverseSuperMatrix", 
    "parity", 
    "superMatrixGenerator", 
    "superMatrixParity", 
    "superRing", 
    "superTrace", 
    
   -- Symbols
    "supermatrix"
}

--------------------
-- SuperRing (Super commutative ring) 
--------------------

superRing = method();
superRing (PolynomialRing, PolynomialRing) := (R1, R2) -> (
    inverseVariable := symbol inverseVariable;
    R11 := (coefficientRing R1)[R1_0..R1_(#gens R1-1), inverseVariable_0..inverseVariable_(#gens R1-1)];
    R11 = R11/apply(0..(#gens R1-1), i -> sub(R1_i, R11)*inverseVariable_i-1);
    w := (for i to (#gens R2)-1 list (0))|toList(0..(#gens R2-1));
    R22 := (coefficientRing R2)[R2_0..R2_(#gens R2-1), MonomialOrder=>{Weights => w, Lex}, SkewCommutative=>true];
    R11**R22
)
 
--------------------
-- SuperMatrix
--------------------

SuperMatrix = new Type of MutableHashTable;
superMatrixGenerator = method();
superMatrixGenerator (Matrix, Matrix, Matrix, Matrix) := (M00,M01,M10,M11)-> (
    new SuperMatrix from {
        supermatrix => matrix{{M00,M01},{M10,M11}}
    }
)

TEST ///
M1 = matrix {{1, 2}, {5, 6}, {9, 10}};
M2 = matrix {{3, 4}, {7, 8}, {11, 12}};
M3 = matrix {{13, 14}, {17, 18}};
M4 = matrix {{15, 16}, {19, 20}};
G = superMatrixGenerator(M1,M2,M3,M4)
assert(G.supermatrix == matrix {{1, 2, 3, 4}, {5, 6, 7, 8}, {9, 10, 11, 12}, {13, 14, 15, 16}, {17, 18, 19, 20}})
///

--------------------
-- parity  
--------------------

parity = method();
parity (RingElement, Ring, List) := (f, R, oddNumberList) -> (
    e := exponents f;
    l := {};
    for i from 0 to (#gens R-1) do (
        for j from 0 to #oddNumberList-1 do (
            if R_(i) == oddNumberList_(j) then (
                l = join(l,{i})
            )
        )
    );
    d := 0;
    countEvenNumber := 0;
    for i from 0 to (#e-1) do (
        if (d%2) == 0 then
             countEvenNumber = countEvenNumber+1;
        d = 0;
        for j from 0 to #l-1 do (
            if (e_i)_(l_j) == 1 then (
                d = d+1
            )
        )
    );
    d = 0;
    for j from 0 to #l-1 do (
        if (e_(#e-1))_(l_j) == 1 then (
            d = d+1
        )
    ); 
    
    if (d%2) == 0 then countEvenNumber = countEvenNumber+1;

if countEvenNumber == #e+1 then 0 
    else if countEvenNumber == 1 then 1 
    else -1
)

parity (Number, Ring, List) := (f, R, oddNumberList) -> (
    0
)

TEST ///
R1 = QQ[x_0..x_3];
R2 = QQ[z_0, z_1];
R = superRing(R1, R2);
a = {z_0, z_1} ;
g = x_1*x_2*x_3+4;
f = x_1*x_2*x_3+x_1*z_0+z_1*z_0-4*x_2*z_1*z_0+4;
h = z_0+z_0*x_0+z_1;
assert(parity(f, R, a) == -1)
assert(parity(g, R, a) == 0)
assert(parity(h, R, a) == 1)
assert(parity(1+2.5*ii, R, a) == 0)
///

--------------------
-- superMatrixParity
--------------------

superMatrixParity = method();
superMatrixParity(SuperMatrix, Ring, List) := (SM, R1, a) -> (
    m1 := 0;
    m2 := 0;
    m3 := 0;
    m4 := 0;
    Minor00 := SM.supermatrix^[0]_[0];
    Minor01 := SM.supermatrix^[0]_[1];
    Minor10 := SM.supermatrix^[1]_[0];
    Minor11 := SM.supermatrix^[1]_[1];
    r1 := numgens target Minor00;
    r2 := numgens target Minor10;
    c1 := numgens source Minor00;
    c2 := numgens source Minor01;
    fij := symbol fij;
    if isSkewCommutative(R1) == true then (
        count1 := 0;
        count11 := 0;
        for i from 0 to (r1-1) do
            for j from 0 to (c1-1) do (
                fij = Minor00_(i, j);
                if fij == 0 then count1 = count1 
                else if (parity(fij, R1, a) == -1) then (count11 = count11+1) 
                else if (parity(fij, R1, a) == 1) then count1 = count1+1
                else if (parity(fij, R1, a) == 0) then count1 = count1
            );
        if count11=!=0 then (return-1) 
        else if count1 == 0 then m1= 0 
        else m1=1;    
        count2 := 0;
        count22 := 0;
        for i from 0 to (r1-1) do 
            for j from 0 to (c2-1) do (
                fij = Minor01_(i, j);
                if fij == 0 then count2 = count2 
                else if (parity(fij, R1, a) == -1) then (count22 = count22+1) 
                else if (parity(fij, R1, a) == 1)then count2 = count2+1
                else if (parity(fij, R1, a) == 0) then count2 = count2
            );
        if count22=!=0 then (return-1) 
        else if count2 == 0 then m2=0 
        else m2=1;
        count3 := 0;
        count33 := 0;
        for i from 0 to (r2-1) do 
            for j from 0 to (c1-1) do (
                fij = Minor10_(i, j);
                if fij == 0 then count3 = count3 
                else if (parity(fij, R1, a) == -1) then (cout33 := count33+1)
                else if (parity(fij, R1, a) == 1)then count3 = count3+1
                else if (parity(fij, R1, a) == 0) then count3 = count3
            );
        if count33=!=0 then (return-1) 
        else if count3 == 0 then m3=0 
        else m3=1;
        count4 := 0;
        count44 := 0;
        for i from 0 to (r2-1) do 
            for j from 0 to (c2-1) do (
                fij = Minor11_(i, j);
                if fij == 0 then count4 = count4 
                else if (parity(fij, R1, a) == -1) then (cout44 := count44+1) 
                else if (parity(fij, R1, a) == 1)then count4 = count4+1
                else if (parity(fij, R1, a) == 0) then count4 = count4
            );
        if count44=!=0 then (return-1) 
        else if count4 == 0 then m4=0 
        else m4=1;
        R2 := coefficientRing R1;
        if (isSkewCommutative(R2) == true) then (
            if (m1 == 0 and m4 == 0 and m2 == 1 and m3 == 1) then (return 0)
            else if (m1 == 1 and m4 == 1 and m2 == 0 and m3 == 0) then (return 1) 
            else (return-1)
        )
        else (
            if (m1 == 0 and m4 == 0 and Minor01 == 0 and Minor10 == 0) then  (return 0)
            else if (Minor00 == 0 and Minor11 == 0 and m2 == 0 and m3 == 0) then (return 1)
            else (return-1)
        )
    )  
    else (error "Ring should be a superRing")
)

TEST///
R1 = QQ[x_0..x_3];
R2 = QQ[z_0..z_2];
R = superRing(R1, R2);
D1 = matrix{{x_0, x_1}, {x_2, x_3}};
D2 = matrix{{z_0, z_1}, {x_0*z_0, x_1*z_1}};
D3 = matrix{{z_2*x_3, z_1}, {z_0, z_2*x_2}};
D4 = matrix{{x_1, x_3}, {x_0, x_2+x_3}};
SD = superMatrixGenerator(D1, D2, D3, D4);
assert(superMatrixParity(SD, R, {z_0, z_1, z_2}) == -1)

P1 = matrix{{0, 0}, {0, 0}};
P2 = matrix{{x_0, x_1}, {x_2, x_3}};
P3 = matrix{{x_1, x_2}, {x_0, x_1}};
P4 = matrix{{0, 0}, {0, 0}};
SP = superMatrixGenerator(P1, P2, P3, P4);
SS = SP.supermatrix;
assert(superMatrixParity(SP, R, {z_0, z_1, z_2}) == 1)

T1 = R[n_0..n_3];
T2 = R[e_0..e_3];
T = superRing(T1, T2);
M1 = matrix{{n_0, n_1}, {n_2, n_3}};
M2 = matrix{{e_0, e_1}, {n_0*e_0, n_1*e_1}};
M3 = matrix{{e_3*n_3, e_1}, {e_0, e_2*n_2}};
M4 = matrix{{n_1, n_3}, {n_0, n_2+n_3}};
SM = superMatrixGenerator(M1, M2, M3, M4);
assert(superMatrixParity(SM, T, {e_0, e_1, e_2, e_3}) == 0)

E1 = matrix{{e_0, n_1}, {n_2, n_3}};
E2 = matrix{{e_0, e_1}, {n_0+e_0, n_1*e_1}};
E3 = matrix{{e_3*n_3, e_1}, {e_0, e_2*n_2}};
E4 = matrix{{n_1, n_3}, {n_0, n_2+n_3}};
G = superMatrixGenerator(E1, E2, E3, E4);
assert(superMatrixParity(G, T, {e_0, e_1, e_2, e_3}) == -1)
///

--------------------
-- Supertrace           
-------------------- 
 
superTrace = method ();
superTrace (SuperMatrix, Ring, List)  := (SM, R1, a) -> (
    Minor00 := SM.supermatrix^[0]_[0];
    Minor11 := SM.supermatrix^[1]_[1];
    if (superMatrixParity(SM, R1, a)=!=-1) then (
        par := superMatrixParity(SM, R1, a);
        trace Minor00-(-1)^par*trace Minor11
    )
    else error "SuperMatrix is not superhomogeneous"
)

TEST ///
R1 = QQ[x_0..x_3];
R2 = QQ[z_0..z_2];
R = superRing(R1, R2);
P1 = matrix{{x_0, x_1}, {x_2, x_3}};
P2 = matrix{{0, 0}, {0, 0}};
P3 = matrix{{0, 0}, {0, 0}};
P4 = matrix{{x_1, x_2}, {x_0, x_1}};
SP = superMatrixGenerator(P1, P2, P3, P4);
assert(superTrace(SP, R, {z_0, z_1}) == x_0-2*x_1+x_3)

R1 = QQ[x_0..x_3]
R2 = QQ[z_0..z_2]
R = superRing(R1, R2)
T1 = R[n_0..n_3];
T2 = R[e_0..e_3];
T = superRing(T1, T2);
M1 = matrix{{n_0, n_1}, {n_2, n_3}};
M2 = matrix{{e_0, e_1}, {n_0*e_0, n_1*e_1}};
M3 = matrix{{e_3*n_3, e_1}, {e_0, e_2*n_2}};
M4 = matrix{{n_1, n_3}, {n_0, n_2+n_3}};
SM = superMatrixGenerator(M1, M2, M3, M4);
a = {e_0, e_1, e_2, e_3};
assert(superTrace(SM, T, a) == n_0-n_1-n_2)
///

--------------------
--berezinian
--------------------

berezinian = method();
berezinian (SuperMatrix, Ring) := (SM, R1) -> (
    Minor00 := SM.supermatrix^[0]_[0];
    Minor01 := SM.supermatrix^[0]_[1];
    Minor10 := SM.supermatrix^[1]_[0];
    Minor11 := SM.supermatrix^[1]_[1];
    SM1 := sub(Minor00, R1);
    SM2 := sub(Minor11, R1);
    Prod1 := Minor11-Minor10*inverse(SM1)*Minor01;
    Prod2 := sub(Prod1, R1);
    if numRows Minor00 =!= numColumns Minor00 then error "expected a square matrix";
    if numRows Minor11 =!= numColumns Minor11 then error "expected a square matrix";
    if det(Minor11) =!= 0 then det(inverse(SM2))*det(Minor00-Minor01*inverse(SM2)*Minor10)
    else if (det(Minor00) =!= 0 and det(Minor11-Minor01*inverse(SM1)*Minor10) =!= 0) then det(Minor00)*det(inverse(Prod2))
    else error "At least one of the diagonal blocks should be invertible"
)
 
TEST///
M1 = matrix{{5, 7}, {1, 2}};
M2 = matrix{{1, 2, 3}, {4, 5, 6}};
M3 = matrix{{3, 4}, {5, 6}, {7, 8}};
M4 = matrix{{2, 3, 11}, {4, 5, 6}, {7, 8, 9}};
M5 = sub(M4, QQ);
G = superMatrixGenerator(M1, M2, M3, M4);
assert(berezinian(G, QQ) ==  det(inverse(M5))*det(M1-M2*inverse(M5)*M3))

S1 = matrix{{1, 2}, {3, 4}};
S2 = matrix{{5, 6}, {7, 8}};
S3 = matrix{{9, 10}, {11, 12}};
S4 = matrix{{0, 0}, {0, 0}};
S5 = sub(S1, QQ);
S6 = S4-S3*inverse(S5)*S2;
F = superMatrixGenerator(S1, S2, S3, S4);
assert(berezinian(F, QQ) == det(S1)*det(inverse(S6)))
///

--------------------
--inversesupermatrix
--------------------

inverseSuperMatrix = method();
inverseSuperMatrix (SuperMatrix, Ring) := (SM, R1) -> (
    Minor00 := SM.supermatrix^[0]_[0];
    Minor01 := SM.supermatrix^[0]_[1];
    Minor10 := SM.supermatrix^[1]_[0];
    Minor11 := SM.supermatrix^[1]_[1];
    if numRows Minor00 =!= numColumns Minor00 then error "expected a square matrix";
    if numRows Minor11 =!= numColumns Minor11 then error "expected a square matrix";    
    SM00 := sub(Minor00, R1);
    SM11 := sub(Minor11, R1);
    SM01 := sub(Minor01, R1);
    SM10 := sub(Minor10, R1);
    Prod1 := SM11-SM10*inverse(SM00)*SM01;
    Prod2 := SM00-SM01*inverse(SM11)*SM10;
    Nminor00 := inverse(Prod2);
    Nminor01 :=-inverse(SM11)*SM10*inverse(Prod2);
    Nminor10 :=-inverse(SM00)*SM01*inverse(Prod1);
    Nminor11 := inverse(Prod1);
    if (det(SM00) =!= 0 and det (SM11) =!= 0) then matrix{{Nminor00,Nminor10},{Nminor01,Nminor11}} else error "The SuperMatrix is not invertible"
)

TEST///
M1 = matrix{{5, 7}, {1, 2}};
M2 = matrix{{1, 2, 3}, {4, 5, 6}};
M3 = matrix{{3, 4}, {5, 6}, {7, 8}};
M4 = matrix{{2, 3, 11}, {4, 5, 6}, {7, 8, 9}};
M44 = sub(M4, QQ);
M11 = sub(M1, QQ);
M22 = sub(M2, QQ);
M33 = sub(M3, QQ);
P2 = M44-M33*inverse(M11)*M22;
P1 = M11-M22*inverse(M44)*M33;
N11 = inverse(P1);
N12 = -inverse(M44)*M33*inverse(P1);
N21 = -inverse(M11)*M22*inverse(P2);
N22 = inverse(P2);
G = superMatrixGenerator(M1, M2, M3, M4);
assert(inverseSuperMatrix(G, QQ) == matrix{{N11,N21},{N12,N22}})
///

--------------------

beginDocumentation()

doc ///
Key
    SuperLinearAlgebra
Headline 
    Package for super algebra
Description
    Text
        This Package is to do the computation in superalgebras, or super vector spaces. 
     
        The computations are taken in a superRing, which is a ring with both symmetric and antisymmetric variables. 
     
        To see the definitions and theorems, see Varadarajan, V. S. (2004). "Supersymmetry for Mathematicians:
        An Introduction" (Vol. 11). American Mathematical Soc.
     
Caveat
SeeAlso
///

doc ///
Key
    superRing
    (superRing, PolynomialRing, PolynomialRing)
Headline
    Makes a super ring from two polynomial rings.
Usage
    R = superRing(R1, R2)
Inputs
    R1:PolynomialRing
    R2:PolynomialRing
Outputs
    R:QuotientRing
        which has both invertible and skew symmetric variables, superRing
Description
    Text
        Let $R_1$ and $R_2$ be Two Polynomial rings on different set of variables
        A superRing is a new polynomial ring with three sets of variables. 
        One set comes from $R_1$ and the second one is the inverse of it.
    
        For example, if we have x as a variable in $R_1$, 
        then there is a new variable, say $inverseVariable_0$ which is the inverse of $x$.
        The third set of variables comes from $R_2$.
        We redefine this set to be a set of skew-symmetric variables.
        So superRing of $R_1$ and $R_2$ is a quotient ring, 
        which has both invertible and skew symmetric variables.
        If the coefficient ring is a field, then we get a super algebra.
    Example
        R1=QQ[x_1..x_5]
        R2=QQ[z_1..z_3]
        superRing(R1, R2)
Caveat
SeeAlso
///

doc ///
Key 
    SuperMatrix
    supermatrix
    superMatrixGenerator
    (superMatrixGenerator , Matrix, Matrix, Matrix, Matrix)
Headline
    Makes a super matrix from its four blocks.
Usage
    G = superMatrixGenerator(M1, M2, M3, M4)
Inputs
    M1:Matrix
    M2:Matrix
    M3:Matrix
    M4:Matrix
Outputs
    G:SuperMatrix
Description
    Text
        Let $M_1, M_2, M_3, M_4$ be four matrices. 
        The number of rows in $M_1$ and $M_2$, 
        and those of $M_3$ and $M_4$ should be equal.
        Also, the number of columns of $M_1$ and $M_3$, 
        and those of M_2 and M_4 must be equal.
   
        The idea is to define a (super) Matrix, 
        which can be considered as $p|q\times r|s$ matrix.
        This super Matrix can be a morphism between super
        modules $A^{p|q}$ and $A^{r|s}$ over super algebra $A$.

The function uses four matrices M_1 and M_2, and also M_3 and M_4
	as four blocks of a new matrix, say $\begin{pmatrix}
        M1&M2\\
        M3&M4\end{pmatrix}$.  
     
        The key supermatrix shows the result matrix created as above.
    Example
        M1 = matrix {{1, 2}, {5, 6}, {9, 10}}
        M2 = matrix {{3, 4}, {7, 8}, {11, 12}}
        M3 = matrix {{13, 14}, {17, 18}}
        M4 = matrix {{15, 16}, {19, 20}}
        G = superMatrixGenerator(M1, M2, M3, M4)
        G.supermatrix
Caveat
SeeAlso
///

doc ///
Key 
    superTrace
    (superTrace, SuperMatrix, Ring, List)
Headline
    Super trace of a homogeneous super matrix.
Usage
    P = superTrace(SM, R, L)
Inputs
    SM:SuperMatrix
    R:Ring
        superRing
    L:List
Outputs
    P:QuotientRing
Description
    Text
        Let $A^{p|q}=Ax_1 \oplus \cdots \oplus Ax_p \oplus Ae_1\oplus \cdots \oplus Ae_q$ be a free module over $A$, 
        where $x_i$s are even and $e_j$s are odd generators. A (homogeneous) morphism $T:A^{p|q}\rightarrow A^{r|s}$ has a matrix representation. 
        Denote the matrix by $T$ then we have $T=\begin{pmatrix}
         T1&T2\\
         T3&T4\end{pmatrix}$.  
     
        The super trace of $T$ is defined by $superTrace(T)= Trace(T_1)-(-1)^{p(T)} Trace(T_4)$.
        The inputs of this function are a SuperMatrix, a ring, which should have skew-symmetric variables, and a list, 
        which is the list of skew-symmetric variables that are used in the superMatrixGenerator. 
        In case that the superMatrix is homogeneous, the output is the super trace of the superMatrix.
        
    Example
        R1 = QQ[x_0..x_3];
        R2 = QQ[z_0..z_2];
        R = superRing(R1, R2);
        P1 = matrix{{x_0, x_1}, {x_2, x_3}};
        P2 = matrix{{0, 0}, {0, 0}};
        P3 = matrix{{0, 0}, {0, 0}};
        P4 = matrix{{x_1, x_2}, {x_0, x_1}};
        SP = superMatrixGenerator(P1, P2, P3, P4);
        superTrace(SP, R, {z_0, z_1})
Caveat
SeeAlso
///

doc ///
Key 
    berezinian
    (berezinian, SuperMatrix, Ring)
Headline
    Computes the berezinian of a supermatrix.
Usage
    N = berezinian(G, R)
Inputs
    G:SuperMatrix
    R:Ring
Outputs
    N:Number
Description
    Text
        This function works only when the entries of the even blocks are numbers, and those of odd blocks are formed by odd generators.
        If in a super Matrix, one of the first or the second diagonal blocks is invertible, 
        then we can define the berezinian (as a kind of super Determinant).
        The formula for the berezinian is different base on which block is invertible.
        But it is shown that the two formulas are equivalent if two blocks are invertible.
        If $M=\begin{pmatrix}
        M1&M2\\
        M3&M4\end{pmatrix}$.  is a super Matrix, and
        $M_4$ is invertible, then 
        $Ber(M)= det(M_1-M_2M^{-1}_4M_3) det(M_4)^{-1}$.
  
        If $M_1$ is invertible, then
        $Ber(M) = det(M_4-M_3M_1^{-1}M_2)^{-1} det(M_1)$.
    Example
        M1 = matrix{{5, 7}, {1, 2}}
        M2 = matrix{{1, 2, 3}, {4, 5, 6}}
        M3 = matrix{{3, 4}, {5, 6}, {7, 8}}
        M4 = matrix{{2, 3, 11}, {4, 5, 6}, {7, 8, 9}}
        M5 = sub(M4, QQ)
        G = superMatrixGenerator(M1, M2, M3, M4)
        berezinian(G, QQ)
Caveat
SeeAlso
///

doc ///
Key 
    parity
    (parity, RingElement, Ring, List)
Headline
    parity of an element of a super ring.
Usage
    N = parity(f, R, L)
Inputs
    f:RingElement
    R:Ring
        superRing
    L:List
Outputs
    N:Number
        0 for even, 1 for odd and-1 for Nonhomogeneous
Description
    Text
        Let we have a super algebra (ring), $R=R_0 \oplus R_1$.
        A homogeneous element of $R$ is an element belongs to $R_0$ or $R_1$.
        This function has three outputs,-1 for non-homogeneous, 0 for homogeneous and even, and 1 for homogeneous and odd elements.
    Example
        R1=QQ[x_0..x_4];
        R2=QQ[e_0, e_1];
        R= superRing(R1, R2)
        L={e_0, e_1}
        f=x_1*x_2*x_3+x_1*e_0+e_1*e_0-4*x_2*e_1*e_0+4
        parity(f, R, L)
        g=x_1*x_2*x_3+e_0*e_1+4;
        parity(g, R, L)
Caveat
SeeAlso
    superMatrixParity
///

doc ///
Key 
    inverseSuperMatrix
    (inverseSuperMatrix, SuperMatrix, Ring) 
Headline
    The inverse of a super matrix.
Usage
    N = inverseSuperMatrix(G, R)
Inputs
    G:SuperMatrix
    R:Ring
Outputs
    M:Matrix
Description
    Text
        A super Matrix $M={{M1, M2}, {M3, M4}}$ 
        is invertible, if both the diagonal blocks, $M_1$ and $M_4$ are invertible. 
  
        In this case, the inverse is given by a blocked matrix, 
        $T=\begin{pmatrix}
         T1&T2\\
         T3&T4\end{pmatrix}$, where
        $T_1=(M_1 − M_2M^{-1}_4 M_3)^{-1}$, 
        $T_2=−M^{-1}_1 M_2(M_4 − M_3M^{-1}_1 M_2)^{-1}$, 
        $T_3=−M^{-1}_4 M_3(M_1 − M_2M^{-1}_4 M_3)^{-1}$, and
        $T_4=(M_4 − M_3M^{-1}_1 M_2)^{-1}$.
    Example
        M1 = matrix{{5, 7}, {1, 2}};
        M2 = matrix{{1, 2, 3}, {4, 5, 6}};
        M3 = matrix{{3, 4}, {5, 6}, {7, 8}};
        M4 = matrix{{2, 3, 11}, {4, 5, 6}, {7, 8, 9}};
        G = superMatrixGenerator(M1, M2, M3, M4);
        inverseSuperMatrix(G, QQ)
Caveat
SeeAlso
///

doc ///
Key 
    superMatrixParity
    (superMatrixParity, SuperMatrix, Ring, List) 
Headline
    parity of a super Matrix.
Usage
    N = superMatrixParity(SM, R, L)
Inputs
    SM:SuperMatrix
    R:Ring
        superRing
    L:List
Outputs
    N:Number
        0 for even, 1 for odd and-1 for Nonhomogeneous
Description
   Text
        This function works only when the entries of the even blocks are numbers, and those of odd blocks are formed by odd generators.
        Let $A^{p|q}=Ax_1\oplus \cdots \oplus Ax_p \oplus Ae_1\oplus\cdots\oplus Ae_q$ 
        be a free module over $A$, where $x_i$s are even and $e_j$s are odd generators. 
        A morphism $T:A^{p|q}\rightarrow A^{r|s}$ has a matrix representation. 
        Denote the matrix by $T$ then we have
        $T=\begin{pmatrix}
         T1&T2\\
         T3&T4\end{pmatrix}$. 
     
        The matrix (morphism) $T$ is said to be even (odd) if the blocks $T_1$ and $T_4$
        are even, and $T_2$ and $T_3$ are odd ($T_1$ and $T_4$ are odd, and $T_2$ and $T_3$ are even).
        Note that if $A$ is an algebra, i.e., it doesn't have odd involution, then to 
        have an even (odd) matrix $T$, we should have $T_3=0$ and $T_2=0$ ($T_1=0$ and $T_4=0$).
    Example
        R1 = QQ[x_0..x_3];
        R2 = QQ[z_0..z_2];
        R = superRing(R1, R2);
        D1 = matrix{{x_0, x_1}, {x_2, x_3}};
        D2 = matrix{{z_0, z_1}, {x_0*z_0, x_1*z_1}};
        D3 = matrix{{z_2*x_3, z_1}, {z_0, z_2*x_2}};
        D4 = matrix{{x_1, x_3}, {x_0, x_2+x_3}};
        SM = superMatrixGenerator(D1, D2, D3, D4);
        superMatrixParity(SM, R, {z_0, z_1, z_2})
Caveat
SeeAlso
    parity
///

end