{{{id=1| m = matrix([[1,2], [3,4]]) /// }}} {{{id=2| m^-1 /// [ -2 1] [ 3/2 -1/2] }}} {{{id=3| m.echelon_form() /// [1 0] [0 2] }}} {{{id=7| print m /// [ 0 0 -1] [ 7 -6 -1] [ -1 -11 1] }}} {{{id=4| m = random_matrix(ZZ, 3); show(m) ///
\left(\begin{array}{rrr} 0 & 0 & -1 \\ 7 & -6 & -1 \\ -1 & -11 & 1 \end{array}\right)
}}} {{{id=5| latex(m) /// \left(\begin{array}{rrr} 0 & 0 & -1 \\ 7 & -6 & -1 \\ -1 & -11 & 1 \end{array}\right) }}} {{{id=6| R. = QQ[[]] /// }}} {{{id=10| R /// Power Series Ring in t over Rational Field }}} {{{id=12| show(R) ///
\Bold{Q}[[t]]
}}} {{{id=11| t /// t }}} {{{id=8| f = sqrt(1+t); f /// 1 + 1/2*t - 1/8*t^2 + 1/16*t^3 - 5/128*t^4 + 7/256*t^5 - 21/1024*t^6 + 33/2048*t^7 - 429/32768*t^8 + 715/65536*t^9 - 2431/262144*t^10 + 4199/524288*t^11 - 29393/4194304*t^12 + 52003/8388608*t^13 - 185725/33554432*t^14 + 334305/67108864*t^15 - 9694845/2147483648*t^16 + 17678835/4294967296*t^17 - 64822395/17179869184*t^18 + 119409675/34359738368*t^19 + O(t^20) }}} {{{id=9| latex(f) /// 1 + \frac{1}{2}t - \frac{1}{8}t^{2} + \frac{1}{16}t^{3} - \frac{5}{128}t^{4} + \frac{7}{256}t^{5} - \frac{21}{1024}t^{6} + \frac{33}{2048}t^{7} - \frac{429}{32768}t^{8} + \frac{715}{65536}t^{9} - \frac{2431}{262144}t^{10} + \frac{4199}{524288}t^{11} - \frac{29393}{4194304}t^{12} + \frac{52003}{8388608}t^{13} - \frac{185725}{33554432}t^{14} + \frac{334305}{67108864}t^{15} - \frac{9694845}{2147483648}t^{16} + \frac{17678835}{4294967296}t^{17} - \frac{64822395}{17179869184}t^{18} + \frac{119409675}{34359738368}t^{19} + O(t^{20}) }}} {{{id=13| show(f) ///
1 + \frac{1}{2}t - \frac{1}{8}t^{2} + \frac{1}{16}t^{3} - \frac{5}{128}t^{4} + \frac{7}{256}t^{5} - \frac{21}{1024}t^{6} + \frac{33}{2048}t^{7} - \frac{429}{32768}t^{8} + \frac{715}{65536}t^{9} - \frac{2431}{262144}t^{10} + \frac{4199}{524288}t^{11} - \frac{29393}{4194304}t^{12} + \frac{52003}{8388608}t^{13} - \frac{185725}{33554432}t^{14} + \frac{334305}{67108864}t^{15} - \frac{9694845}{2147483648}t^{16} + \frac{17678835}{4294967296}t^{17} - \frac{64822395}{17179869184}t^{18} + \frac{119409675}{34359738368}t^{19} + O(t^{20})
}}} {{{id=14| sage_input(f.truncate()) /// R. = QQ[] 119409675/34359738368*t^19 - 64822395/17179869184*t^18 + 17678835/4294967296*t^17 - 9694845/2147483648*t^16 + 334305/67108864*t^15 - 185725/33554432*t^14 + 52003/8388608*t^13 - 29393/4194304*t^12 + 4199/524288*t^11 - 2431/262144*t^10 + 715/65536*t^9 - 429/32768*t^8 + 33/2048*t^7 - 21/1024*t^6 + 7/256*t^5 - 5/128*t^4 + 1/16*t^3 - 1/8*t^2 + 1/2*t + 1 }}} {{{id=15| factor(14832) /// 2^4 * 3^2 * 103 }}} {{{id=16| R. = QQbar[] /// }}} {{{id=17| R /// Univariate Polynomial Ring in x over Algebraic Field }}} {{{id=18| a, b = (x^2-2).roots() /// }}} {{{id=21| a[0].minpoly() /// x^2 - 2 }}} {{{id=22| m = random_matrix(ZZ, 5, density=0.2) /// }}} {{{id=23| m /// [ 0 0 1 0 0] [-4 0 0 0 0] [ 0 1 0 0 0] [-1 0 0 0 0] [ 0 1 0 0 0] }}} {{{id=24| b = random_matrix(ZZ, 5, 5) * m /// }}} {{{id=25| x = m.solve_left(b) /// }}} {{{id=26| x*m == b /// True }}} {{{id=27| x = m.solve_right(b) /// }}} {{{id=28| m*x == b /// True }}} {{{id=29| x = m \ b /// }}} {{{id=30| m*x /// (1, 2, 3, 4, 5) }}} {{{id=31| m = random_matrix(ZZ, 3000, density=.001) /// }}} {{{id=32| time m.rank() /// 2652 Time: CPU 17.56 s, Wall: 17.11 s }}} {{{id=33| m = random_matrix(ZZ, 3) /// }}} {{{id=34| show(m.eigenspaces()) ///
\left[\left(a_{0}, \mathrm{RowSpan}_{\Bold{Q}[a_{0}]/(a_{0}^{3} + 4 a_{0}^{2} + 7)}\left(\begin{array}{rrr} 1 & -\frac{1}{3} a_{0}^{2} - \frac{2}{3} a_{0} + \frac{7}{3} & \frac{1}{3} a_{0}^{2} + \frac{5}{3} a_{0} + \frac{2}{3} \end{array}\right)\right)\right]
}}} {{{id=35| m = random_matrix(GF(17), 3) /// }}} {{{id=36| m.eigenspaces() /// [ (15, Vector space of degree 3 and dimension 1 over Finite Field of size 17 User basis matrix: [1 7 6]), (13, Vector space of degree 3 and dimension 1 over Finite Field of size 17 User basis matrix: [ 1 13 12]), (9, Vector space of degree 3 and dimension 1 over Finite Field of size 17 User basis matrix: [ 1 11 5]) ] }}} {{{id=37| m = random_matrix(QQ, 10) /// }}} {{{id=38| m.minpoly() /// x^10 - x^9 - 11*x^8 + 73/8*x^7 - 100*x^6 - 541/32*x^5 + 45189/64*x^4 + 2135/32*x^3 - 220523/128*x^2 - 670739/64*x - 959713/128 }}} {{{id=39| R = Frac(Qp(7)['x,y,z']) /// }}} {{{id=41| m = matrix(R, [[1,2],[3,R.gen()]]); show(m) ///
\left(\begin{array}{rr} 1 + O(7^{20}) & 2 + O(7^{20}) \\ 3 + O(7^{20}) & x \end{array}\right)
}}} {{{id=42| show(m^4) ///
\left(\begin{array}{rr} (6 + O(7^{20})) x^{2} + (5 + 7 + O(7^{20})) x + 6 + 7^{2} + O(7^{20}) & (2 + O(7^{20})) x^{3} + (2 + O(7^{20})) x^{2} + (5 + 3 \cdot 7 + O(7^{20})) x + 5 + 3 \cdot 7 + O(7^{20}) \\ (3 + O(7^{20})) x^{3} + (3 + O(7^{20})) x^{2} + (4 + 5 \cdot 7 + O(7^{20})) x + 4 + 5 \cdot 7 + O(7^{20}) & x^{4} + (4 + 2 \cdot 7 + O(7^{20})) x^{2} + (5 + 7 + O(7^{20})) x + 6 \cdot 7 + O(7^{20}) \end{array}\right)
}}} {{{id=43| show(m.echelon.261_form()) ///
\left(\begin{array}{rr} 1 + O(7^{20}) & 0 \\ 0 & \frac{x + 1 + 6 \cdot 7 + 6 \cdot 7^{2} + 6 \cdot 7^{3} + 6 \cdot 7^{4} + 6 \cdot 7^{5} + 6 \cdot 7^{6} + 6 \cdot 7^{7} + 6 \cdot 7^{8} + 6 \cdot 7^{9} + 6 \cdot 7^{10} + 6 \cdot 7^{11} + 6 \cdot 7^{12} + 6 \cdot 7^{13} + 6 \cdot 7^{14} + 6 \cdot 7^{15} + 6 \cdot 7^{16} + 6 \cdot 7^{17} + 6 \cdot 7^{18} + 6 \cdot 7^{19} + O(7^{20})}{x + 1 + 6 \cdot 7 + 6 \cdot 7^{2} + 6 \cdot 7^{3} + 6 \cdot 7^{4} + 6 \cdot 7^{5} + 6 \cdot 7^{6} + 6 \cdot 7^{7} + 6 \cdot 7^{8} + 6 \cdot 7^{9} + 6 \cdot 7^{10} + 6 \cdot 7^{11} + 6 \cdot 7^{12} + 6 \cdot 7^{13} + 6 \cdot 7^{14} + 6 \cdot 7^{15} + 6 \cdot 7^{16} + 6 \cdot 7^{17} + 6 \cdot 7^{18} + 6 \cdot 7^{19} + O(7^{20})} \end{array}\right)
}}} {{{id=44| show(m.echelon_form()) ///
\left(\begin{array}{rr} 1 + O(7^{20}) & 0 \\ 0 & \frac{x + 1 + 6 \cdot 7 + 6 \cdot 7^{2} + 6 \cdot 7^{3} + 6 \cdot 7^{4} + 6 \cdot 7^{5} + 6 \cdot 7^{6} + 6 \cdot 7^{7} + 6 \cdot 7^{8} + 6 \cdot 7^{9} + 6 \cdot 7^{10} + 6 \cdot 7^{11} + 6 \cdot 7^{12} + 6 \cdot 7^{13} + 6 \cdot 7^{14} + 6 \cdot 7^{15} + 6 \cdot 7^{16} + 6 \cdot 7^{17} + 6 \cdot 7^{18} + 6 \cdot 7^{19} + O(7^{20})}{x + 1 + 6 \cdot 7 + 6 \cdot 7^{2} + 6 \cdot 7^{3} + 6 \cdot 7^{4} + 6 \cdot 7^{5} + 6 \cdot 7^{6} + 6 \cdot 7^{7} + 6 \cdot 7^{8} + 6 \cdot 7^{9} + 6 \cdot 7^{10} + 6 \cdot 7^{11} + 6 \cdot 7^{12} + 6 \cdot 7^{13} + 6 \cdot 7^{14} + 6 \cdot 7^{15} + 6 \cdot 7^{16} + 6 \cdot 7^{17} + 6 \cdot 7^{18} + 6 \cdot 7^{19} + O(7^{20})} \end{array}\right)
}}} {{{id=45| m = random_matrix(ZZ, 5) /// }}} {{{id=46| show(m.smith_form()) ///
\left(\left(\begin{array}{rrrrr} 1 & 0 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 & 0 \\ 0 & 0 & 1 & 0 & 0 \\ 0 & 0 & 0 & 3 & 0 \\ 0 & 0 & 0 & 0 & 375 \end{array}\right), \left(\begin{array}{rrrrr} 0 & 0 & 0 & 0 & 1 \\ 0 & 0 & 0 & 1 & 0 \\ 0 & 0 & 1 & 0 & 0 \\ 0 & 1 & 0 & -2 & -2 \\ 1 & -83 & 0 & -21 & 42 \end{array}\right), \left(\begin{array}{rrrrr} 59 & 89 & -2 & 118 & 177 \\ -30 & -46 & 1 & -61 & -94 \\ -29 & -44 & 1 & -58 & -87 \\ 0 & 0 & 0 & 0 & 1 \\ -2 & -3 & 0 & -4 & -6 \end{array}\right)\right)
}}} {{{id=47| m = random_matrix(ZZ, 150) time h = (m.hermite_form()) /// Time: CPU 1.13 s, Wall: 1.36 s }}} {{{id=48| h[-1,-1] /// 8375677550724595192675641077288237457048522483917710157652778465125657401802410520334228193073121691278932506332980975346116095028405721348080390321223130549269468876005491649019466447132328083540300918478697656344292037964238876241671235887400940698709674611150640552770635581963126609487375496376019485012273435032381048445132650095 }}} {{{id=49| m = random_matrix(ZZ, 5); m /// [ 2 1 23 -1 0] [ 0 -1 1 0 -8] [ 4 -2 7 4 3] [ 2 -3 0 -1 2] [ 0 -1 0 -1 1] }}} {{{id=50| m.column(2) /// (0, 0, 1) }}} {{{id=51| m[:2,:3] /// [ 2 1 23] [ 0 -1 1] }}} {{{id=52| L = range(10) L /// [0, 1, 2, 3, 4, 5, 6, 7, 8, 9] }}} {{{id=53| L[:4] /// [0, 1, 2, 3] }}} {{{id=54| L[2:4] /// [2, 3] }}} {{{id=55| V = QQ^3 V /// Vector space of dimension 3 over Rational Field }}} {{{id=60| V.basis() /// [ (1, 0, 0), (0, 1, 0), (0, 0, 1) ] }}} {{{id=59| V.1 /// (0, 1, 0) }}} {{{id=56| W = span(QQ, (V.gen(0), V.1)) /// }}} {{{id=57| V/W /// Vector space quotient V/W of dimension 1 over Rational Field where V: Vector space of dimension 3 over Rational Field W: Vector space of degree 3 and dimension 2 over Rational Field Basis matrix: [1 0 0] [0 1 0] }}} {{{id=58| /// }}}