    /**************************************************************************\

    MODULE: mat_ZZ_pE

    SUMMARY:

    Defines the class mat_ZZ_pE.

    \**************************************************************************/

    #include <NTL/matrix.h>
    #include <NTL/vec_vec_ZZ_pE.h>

    typedef Mat<ZZ_pE> mat_ZZ_pE; // backward compatibility

    void add(mat_ZZ_pE& X, const mat_ZZ_pE& A, const mat_ZZ_pE& B);
    // X = A + B

    void sub(mat_ZZ_pE& X, const mat_ZZ_pE& A, const mat_ZZ_pE& B);
    // X = A - B

    void negate(mat_ZZ_pE& X, const mat_ZZ_pE& A);
    // X = - A

    void mul(mat_ZZ_pE& X, const mat_ZZ_pE& A, const mat_ZZ_pE& B);
    // X = A * B

    void mul(vec_ZZ_pE& x, const mat_ZZ_pE& A, const vec_ZZ_pE& b);
    // x = A * b

    void mul(vec_ZZ_pE& x, const vec_ZZ_pE& a, const mat_ZZ_pE& B);
    // x = a * B

    void mul(mat_ZZ_pE& X, const mat_ZZ_pE& A, const ZZ_pE& b);
    void mul(mat_ZZ_pE& X, const mat_ZZ_pE& A, const ZZ_p& b);
    void mul(mat_ZZ_pE& X, const mat_ZZ_pE& A, long b);
    // X = A * b

    void mul(mat_ZZ_pE& X, const ZZ_pE& a, const mat_ZZ_pE& B);
    void mul(mat_ZZ_pE& X, const ZZ_p& a, const mat_ZZ_pE& B);
    void mul(mat_ZZ_pE& X, long a, const mat_ZZ_pE& B);
    // X = a * B

    void determinant(ZZ_pE& d, const mat_ZZ_pE& A);
    ZZ_pE determinant(const mat_ZZ_pE& a);
    // d = determinant(A)

    void transpose(mat_ZZ_pE& X, const mat_ZZ_pE& A);
    mat_ZZ_pE transpose(const mat_ZZ_pE& A);
    // X = transpose of A

    void solve(ZZ_pE& d, vec_ZZ_pE& x, const mat_ZZ_pE& A, const vec_ZZ_pE& b);
    // A is an n x n matrix, b is a length n vector.  Computes d = determinant(A).
    // If d != 0, solves x*A = b.

    void solve(ZZ_pE& d, const mat_ZZ_pE& A, vec_ZZ_pE& x, const vec_ZZ_pE& b);
    // A is an n x n matrix, b is a length n vector.  Computes d = determinant(A).
    // If d != 0, solves A*x = b (so x and b are treated as a column vectors).

    void inv(ZZ_pE& d, mat_ZZ_pE& X, const mat_ZZ_pE& A);
    // A is an n x n matrix.  Computes d = determinant(A).  If d != 0,
    // computes X = A^{-1}.

    void sqr(mat_ZZ_pE& X, const mat_ZZ_pE& A);
    mat_ZZ_pE sqr(const mat_ZZ_pE& A);
    // X = A*A

    void inv(mat_ZZ_pE& X, const mat_ZZ_pE& A);
    mat_ZZ_pE inv(const mat_ZZ_pE& A);
    // X = A^{-1}; error is raised if A is  singular

    void power(mat_ZZ_pE& X, const mat_ZZ_pE& A, const ZZ& e);
    mat_ZZ_pE power(const mat_ZZ_pE& A, const ZZ& e);

    void power(mat_ZZ_pE& X, const mat_ZZ_pE& A, long e);
    mat_ZZ_pE power(const mat_ZZ_pE& A, long e);
    // X = A^e; e may be negative (in which case A must be nonsingular).

    void ident(mat_ZZ_pE& X, long n);
    mat_ZZ_pE ident_mat_ZZ_pE(long n);
    // X = n x n identity matrix

    long IsIdent(const mat_ZZ_pE& A, long n);
    // test if A is the n x n identity matrix

    void diag(mat_ZZ_pE& X, long n, const ZZ_pE& d);
    mat_ZZ_pE diag(long n, const ZZ_pE& d);
    // X = n x n diagonal matrix with d on diagonal

    long IsDiag(const mat_ZZ_pE& A, long n, const ZZ_pE& d);
    // test if X is an  n x n diagonal matrix with d on diagonal

    void random(mat_ZZ_pE& x, long n, long m);  // x = random n x m matrix
    mat_ZZ_pE random_mat_ZZ_pE(long n, long m);

    long gauss(mat_ZZ_pE& M);
    long gauss(mat_ZZ_pE& M, long w);
    // Performs unitary row operations so as to bring M into row echelon
    // form.  If the optional argument w is supplied, stops when first w
    // columns are in echelon form.  The return value is the rank (or the
    // rank of the first w columns).

    void image(mat_ZZ_pE& X, const mat_ZZ_pE& A);
    // The rows of X are computed as basis of A's row space.  X is is row
    // echelon form

    void kernel(mat_ZZ_pE& X, const mat_ZZ_pE& A);
    // Computes a basis for the kernel of the map x -> x*A. where x is a
    // row vector.

    // miscellaneous:

    void clear(mat_ZZ_pE& a);
    // x = 0 (dimension unchanged)

    long IsZero(const mat_ZZ_pE& a);
    // test if a is the zero matrix (any dimension)

    // operator notation:

    mat_ZZ_pE operator+(const mat_ZZ_pE& a, const mat_ZZ_pE& b);
    mat_ZZ_pE operator-(const mat_ZZ_pE& a, const mat_ZZ_pE& b);
    mat_ZZ_pE operator*(const mat_ZZ_pE& a, const mat_ZZ_pE& b);

    mat_ZZ_pE operator-(const mat_ZZ_pE& a);

    // matrix/scalar multiplication:

    mat_ZZ_pE operator*(const mat_ZZ_pE& a, const ZZ_pE& b);
    mat_ZZ_pE operator*(const mat_ZZ_pE& a, const ZZ_p& b);
    mat_ZZ_pE operator*(const mat_ZZ_pE& a, long b);

    mat_ZZ_pE operator*(const ZZ_pE& a, const mat_ZZ_pE& b);
    mat_ZZ_pE operator*(const ZZ_p& a, const mat_ZZ_pE& b);
    mat_ZZ_pE operator*(long a, const mat_ZZ_pE& b);

    // matrix/vector multiplication:

    vec_ZZ_pE operator*(const mat_ZZ_pE& a, const vec_ZZ_pE& b);

    vec_ZZ_pE operator*(const vec_ZZ_pE& a, const mat_ZZ_pE& b);

    // assignment operator notation:

    mat_ZZ_pE& operator+=(mat_ZZ_pE& x, const mat_ZZ_pE& a);
    mat_ZZ_pE& operator-=(mat_ZZ_pE& x, const mat_ZZ_pE& a);
    mat_ZZ_pE& operator*=(mat_ZZ_pE& x, const mat_ZZ_pE& a);

    mat_ZZ_pE& operator*=(mat_ZZ_pE& x, const ZZ_pE& a);
    mat_ZZ_pE& operator*=(mat_ZZ_pE& x, const ZZ_p& a);
    mat_ZZ_pE& operator*=(mat_ZZ_pE& x, long a);

    vec_ZZ_pE& operator*=(vec_ZZ_pE& x, const mat_ZZ_pE& a);