线性代数¶

向量空间¶

这个 VectorSpace 命令创建一个向量空间类，从中可以创建子空间。注：Sage计算的基础是“行减少”。

sage: V = VectorSpace(GF(2),8)
sage: S = V.subspace([V([1,1,0,0,0,0,0,0]),V([1,0,0,0,0,1,1,0])])
sage: S.basis()
[
(1, 0, 0, 0, 0, 1, 1, 0),
(0, 1, 0, 0, 0, 1, 1, 0)
]
sage: S.dimension()
2

矩阵幂¶

在Sage中如何计算矩阵幂？下面的示例说明了语法。

sage: R = IntegerModRing(51)
sage: M = MatrixSpace(R,3,3)
sage: A = M([1,2,3, 4,5,6, 7,8,9])
sage: A^1000*A^1007
<BLANKLINE>
[ 3  3  3]
[18  0 33]
[33 48 12]
sage: A^2007
<BLANKLINE>
[ 3  3  3]
[18  0 33]
[33 48 12]

籽粒¶

通过对矩阵对象应用核方法来计算核。下面的例子说明了语法。

sage: M = MatrixSpace(IntegerRing(),4,2)(range(8))
sage: M.kernel()
Free module of degree 4 and rank 2 over Integer Ring
Echelon basis matrix:
[ 1  0 -3  2]
[ 0  1 -2  1]

一个超维核 $\QQ$ ：

sage: A = MatrixSpace(RationalField(),3)(range(9))
sage: A.kernel()
Vector space of degree 3 and dimension 1 over Rational Field
Basis matrix:
[ 1 -2  1]

微不足道的内核：

sage: A = MatrixSpace(RationalField(),2)([1,2,3,4])
sage: A.kernel()
Vector space of degree 2 and dimension 0 over Rational Field
Basis matrix:
[]
sage: M = MatrixSpace(RationalField(),0,2)(0)
sage: M
[]
sage: M.kernel()
Vector space of degree 0 and dimension 0 over Rational Field
Basis matrix:
[]
sage: M = MatrixSpace(RationalField(),2,0)(0)
sage: M.kernel()
Vector space of degree 2 and dimension 2 over Rational Field
Basis matrix:
[1 0]
[0 1]

零矩阵的核：

sage: A = MatrixSpace(RationalField(),2)(0)
sage: A.kernel()
Vector space of degree 2 and dimension 2 over Rational Field
Basis matrix:
[1 0]
[0 1]

非方阵的核：

sage: A = MatrixSpace(RationalField(),3,2)(range(6))
sage: A.kernel()
Vector space of degree 3 and dimension 1 over Rational Field
Basis matrix:
[ 1 -2  1]

分圆域上矩阵的二维核：

sage: K = CyclotomicField(12); a = K.gen()
sage: M = MatrixSpace(K,4,2)([1,-1, 0,-2, 0,-a^2-1, 0,a^2-1])
sage: M
[             1            -1]
[             0            -2]
[             0 -zeta12^2 - 1]
[             0  zeta12^2 - 1]
sage: M.kernel()
Vector space of degree 4 and dimension 2 over Cyclotomic Field of order 12
 and degree 4
Basis matrix:
[               0                1                0     -2*zeta12^2]
[               0                0                1 -2*zeta12^2 + 1]

复杂基域上的非平凡核。

sage: K = FractionField(PolynomialRing(RationalField(),2,'x'))
sage: M = MatrixSpace(K, 2)([[K.gen(1),K.gen(0)], [K.gen(1), K.gen(0)]])
sage: M
[x1 x0]
[x1 x0]
sage: M.kernel()
Vector space of degree 2 and dimension 1 over Fraction Field of Multivariate
Polynomial Ring in x0, x1 over Rational Field
Basis matrix:
 [ 1 -1]

整数矩阵的其他方法有 elementary_divisors ， smith_form （对于史密斯范式）， echelon_form 对于Hermite范式， frobenius 对于Frobenius规范型（有理规范型）。

对于一个域上的矩阵有很多方法，比如 $\QQ$ 或有限域： row_span ， nullity ， transpose ， swap_rows ， matrix_from_columns ， matrix_from_rows ，以及许多其他的。

查看文件 matrix.py 更多详情。

特征向量和特征值¶

如何使用Sage计算特征值和特征向量？

Sage具有计算特征值、左右特征向量和特征空间的全套函数。如果我们的矩阵是 $A$ 然后 eigenmatrix_right （RESP） eightmatrix_left )命令也给出矩阵 $D$ 和 $P$ 这样的话 $AP=PD$ （RESP） $PA=DP$ ）

sage: A = matrix(QQ, [[1,1,0],[0,2,0],[0,0,3]])
sage: A
[1 1 0]
[0 2 0]
[0 0 3]
sage: A.eigenvalues()
[3, 2, 1]
sage: A.eigenvectors_right()
[(3, [
(0, 0, 1)
], 1), (2, [
(1, 1, 0)
], 1), (1, [
(1, 0, 0)
], 1)]

sage: A.eigenspaces_right()
[
(3, Vector space of degree 3 and dimension 1 over Rational Field
User basis matrix:
[0 0 1]),
(2, Vector space of degree 3 and dimension 1 over Rational Field
User basis matrix:
[1 1 0]),
(1, Vector space of degree 3 and dimension 1 over Rational Field
User basis matrix:
[1 0 0])
]

sage: D, P = A.eigenmatrix_right()
sage: D
[3 0 0]
[0 2 0]
[0 0 1]
sage: P
[0 1 1]
[0 1 0]
[1 0 0]
sage: A*P == P*D
True

For eigenvalues outside the fraction field of the base ring of the matrix, you can choose to have all the eigenspaces output when the algebraic closure of the field is implemented, such as the algebraic numbers, QQbar. Or you may request just a single eigenspace for each irreducible factor of the characteristic polynomial, since the others may be formed through Galois conjugation. The eigenvalues of the matrix below are $pmsqrt{3}$ and we exhibit each possible output.

而且，目前Sage没有实现多精度的数值特征值和特征向量，因此从 CC 或 RR 可能会给出不准确和无意义的结果（也会打印警告）。具有浮点项的矩阵的特征值和特征向量（上 CDF 和 RDF )可以通过“特征矩阵”命令获得。:

sage: MS = MatrixSpace(QQ, 2, 2)
sage: A = MS([1,-4,1, -1])
sage: A.eigenspaces_left(format='all')
[
(-1.732050807568878?*I, Vector space of degree 2 and dimension 1 over Algebraic Field
User basis matrix:
[                        1 -1 - 1.732050807568878?*I]),
(1.732050807568878?*I, Vector space of degree 2 and dimension 1 over Algebraic Field
User basis matrix:
[                        1 -1 + 1.732050807568878?*I])
]
sage: A.eigenspaces_left(format='galois')
[
(a0, Vector space of degree 2 and dimension 1 over Number Field in a0 with defining polynomial x^2 + 3
User basis matrix:
[     1 a0 - 1])
]

另一种方法是使用与Maxima的接口：

sage: A = maxima("matrix ([1, -4], [1, -1])")
sage: eig = A.eigenvectors()
sage: eig
[[[-sqrt(3)*%i,sqrt(3)*%i],[1,1]],[[[1,(sqrt(3)*%i+1)/4]],[[1,-(sqrt(3)*%i-1)/4]]]]

这告诉我们 $\vec{{v}}_1 = [1,(\sqrt{{3}}i + 1)/4]$ 是的特征向量 $\lambda_1 = - \sqrt{{3}}i$ （以多重数1出现）和 $\vec{{v}}_2 = [1,(-\sqrt{{3}}i + 1)/4]$ 是的特征向量 $\lambda_2 = \sqrt{{3}}i$ （这也发生在多重性一的情况下）。

这里还有两个例子：

sage: A = maxima("matrix ([11, 0, 0], [1, 11, 0], [1, 3, 2])")
sage: A.eigenvectors()
[[[2,11],[1,2]],[[[0,0,1]],[[0,1,1/3]]]]
sage: A = maxima("matrix ([-1, 0, 0], [1, -1, 0], [1, 3, 2])")
sage: A.eigenvectors()
[[[-1,2],[2,1]],[[[0,1,-1]],[[0,0,1]]]]

警告：注意输出的顺序是如何颠倒的，尽管矩阵几乎相同。

最后，您还可以使用Sage的GAP接口来计算“有理”特征值和特征向量：

sage: print(gap.eval("A := [[1,2,3],[4,5,6],[7,8,9]]"))
[ [ 1, 2, 3 ], [ 4, 5, 6 ], [ 7, 8, 9 ] ]
sage: print(gap.eval("v := Eigenvectors( Rationals,A)"))
[ [ 1, -2, 1 ] ]
sage: print(gap.eval("lambda := Eigenvalues( Rationals,A)"))
[ 0 ]

行缩减¶

矩阵的行归约梯队形式如下例所示。

sage: M = MatrixSpace(RationalField(),2,3)
sage: A = M([1,2,3, 4,5,6])
sage: A
[1 2 3]
[4 5 6]
sage: A.parent()
Full MatrixSpace of 2 by 3 dense matrices over Rational Field
sage: A[0,2] = 389
sage: A
[  1   2 389]
[  4   5   6]
sage: A.echelon_form()
[      1       0 -1933/3]
[      0       1  1550/3]

特征多项式¶

特征多项式是求解方阵的一种有效方法。

首先是矩阵 $\ZZ$ ：

sage: A = MatrixSpace(IntegerRing(),2)( [[1,2], [3,4]] )
sage: f = A.charpoly()
sage: f
x^2 - 5*x - 2
sage: f.parent()
Univariate Polynomial Ring in x over Integer Ring

我们计算多项式环上矩阵的特征多项式 $\ZZ[a]$ ：

sage: R = PolynomialRing(IntegerRing(),'a'); a = R.gen()
sage: M = MatrixSpace(R,2)([[a,1], [a,a+1]])
sage: M
[    a     1]
[    a a + 1]
sage: f = M.charpoly()
sage: f
x^2 + (-2*a - 1)*x + a^2
sage: f.parent()
Univariate Polynomial Ring in x over Univariate Polynomial Ring in a over
Integer Ring

sage: M.trace()
2*a + 1
sage: M.determinant()
a^2

我们计算了多元多项式环上矩阵的特征多项式 $\ZZ[u,v]$ ：

sage: R.<u,v> = PolynomialRing(ZZ,2)
sage: A = MatrixSpace(R,2)([u,v,u^2,v^2])
sage: f = A.charpoly(); f
x^2 + (-v^2 - u)*x - u^2*v + u*v^2

区分变量有点困难。为了解决这一问题，我们可能需要将不确定的“Z”重命名为“Z”，我们可以很容易地执行以下操作：

sage: f = A.charpoly('Z'); f
Z^2 + (-v^2 - u)*Z - u^2*v + u*v^2

解线性方程组¶

使用maxima，您可以轻松地求解线性方程：

sage: var('a,b,c')
(a, b, c)
sage: eqn = [a+b*c==1, b-a*c==0, a+b==5]
sage: s = solve(eqn, a,b,c); s
[[a == -1/4*I*sqrt(79) + 11/4, b == 1/4*I*sqrt(79) + 9/4, c == 1/10*I*sqrt(79) + 1/10], [a == 1/4*I*sqrt(79) + 11/4, b == -1/4*I*sqrt(79) + 9/4, c == -1/10*I*sqrt(79) + 1/10]]

你甚至可以在 Latex 中很好地排版解决方案：

sage.: print(latex(s))
...

要通过xdvi在屏幕上显示上述内容，请键入 view(s) .

也可以使用 solve 命令：

sage: var('x,y,z,a')
(x, y, z, a)
sage: eqns = [x + z == y, 2*a*x - y == 2*a^2, y - 2*z == 2]
sage: solve(eqns, x, y, z)
[[x == a + 1, y == 2*a, z == a - 1]]

下面是一个数字纽比的例子：

sage: from numpy import arange, eye, linalg
sage: A = eye(10)       ##   the 10x10 identity matrix
sage: b = arange(1,11)
sage: x = linalg.solve(A,b)

另一种数值求解系统的方法是使用Sage的倍频程接口：

sage: M33 = MatrixSpace(QQ,3,3)
sage: A   = M33([1,2,3,4,5,6,7,8,0])
sage: V3  = VectorSpace(QQ,3)
sage: b   = V3([1,2,3])
sage: octave.solve_linear_system(A,b)    # optional - octave
[-0.333333, 0.666667, 0]