1.1.6. 矩阵¶
参考 Matrices
要在 SymPy 中创建矩阵，请使用 Matrix 对象。通过提供组成矩阵的行向量列表来构建矩阵。例如，构造矩阵
import warnings
from sympy import *
warnings.filterwarnings('ignore')
Matrix([[1, -1], [3, 4], [0, 2]])
为了便于制作列向量，元素列表被认为是列向量。
Matrix([1, 2, 3])
矩阵的操作就像 SymPy 或 Python 中的任何其他对象一样。
M = Matrix([[1, 2, 3], [3, 2, 1]])
N = Matrix([0, 1, 1])
M * N
关于 SymPy 矩阵需要注意的一件重要事情是，与 SymPy 中的所有其他对象不同，它们是可变的。这意味着它们可以就地修改，我们将在下面看到。这样做的缺点是 Matrix 不能用于需要不变性的地方，例如在其他 SymPy 表达式内部或作为字典的键。如果您需要 Matrix 的不可变版本，请使用 ImmutableMatrix。
Matrix 的基本运算¶
shape¶
要获得矩阵的形状，请使用 shape() 函数。
from sympy import shape
M = Matrix([[1, 2, 3], [-2, 0, 4]])
M
shape(M)
(2, 3)
访问行和列¶
要获取矩阵的单个行或列，请使用 row 或 col。例如，M.row(0) 将获得第一行。M.col(-1) 将获得最后一列。
M.row(0), M.col(-1)
(Matrix([[1, 2, 3]]),
Matrix([
[3],
[4]]))
删除和插入行和列¶
要删除行或列，请使用 row_del 或 col_del。这些操作将就地修改矩阵。
M.col_del(0)
M.row_del(1)
要插入行或列，请使用 row_insert 或 col_insert。这些操作没有就地操作。
M
M = M.row_insert(1, Matrix([[0, 4]]))
M
M = M.col_insert(0, Matrix([1, -2]))
M
除非明确说明，否则下面提到的方法不会就地运行。通常，未就地操作的方法将返回一个新的 Matrix，而就地操作的方法将返回 None。
基础方法¶
如上所述，加法和乘法等简单运算只需使用 +、* 和 ** 即可完成。要找到矩阵的逆矩阵，只需 -1 次方即可。
M = Matrix([[1, 3], [-2, 3]])
N = Matrix([[0, 3], [0, 7]])
M + N
M -N
3 * M
M * N
M ** 2
M ** -1
要对矩阵进行转置，请使用 T。
M = Matrix([[1, 2, 3], [4, 5, 6]])
M
M.T
其他方法¶
eye(n) 生成 \(n \times n\) 单位矩阵，zeros(n, m)，ones(n, m) 分别为全零矩阵，全一矩阵。diag 生成对角矩阵：
diag(1, 2, 3)
diag(-1, ones(2, 2), Matrix([5, 7, 5]))
det 计算行列式：
M = Matrix([[1, 0, 1], [2, -1, 3], [4, 3, 2]])
M.det()
rref¶
To put a matrix into reduced row echelon form, use rref. rref returns a tuple of two elements. The first is the reduced row echelon form, and the second is a tuple of indices of the pivot columns.
M = Matrix([[1, 0, 1, 3], [2, 3, 4, 7], [-1, -3, -3, -4]])
M
M.rref()
(Matrix([
[1, 0, 1, 3],
[0, 1, 2/3, 1/3],
[0, 0, 0, 0]]),
(0, 1))
nullspace¶
To find the nullspace of a matrix, use nullspace. nullspace returns a list of column vectors that span the nullspace of the matrix.
M = Matrix([[1, 2, 3, 0, 0], [4, 10, 0, 0, 1]])
M
M.nullspace()
[Matrix([
[-15],
[ 6],
[ 1],
[ 0],
[ 0]]),
Matrix([
[0],
[0],
[0],
[1],
[0]]),
Matrix([
[ 1],
[-1/2],
[ 0],
[ 0],
[ 1]])]
columnspace¶
To find the columnspace of a matrix, use columnspacee. columnspace returns a list of column vectors that span the nullspace of the matrix.
M = Matrix([[1, 1, 2], [2 ,1 , 3], [3 , 1, 4]])
M
M.columnspace()
[Matrix([
[1],
[2],
[3]]),
Matrix([
[1],
[1],
[1]])]
Eigenvalues, Eigenvectors, and Diagonalization¶
To find the eigenvalues of a matrix, use eigenvals. eigenvals returns a dictionary of eigenvalue: algebraic_multiplicity pairs (similar to the output of roots).
M = Matrix([[3, -2, 4, -2], [5, 3, -3, -2], [5, -2, 2, -2], [5, -2, -3, 3]])
M
M.eigenvals()
{3: 1, -2: 1, 5: 2}
This means that M has eigenvalues -2, 3, and 5, and that the eigenvalues -2 and 3 have algebraic multiplicity 1 and that the eigenvalue 5 has algebraic multiplicity 2.
To find the eigenvectors of a matrix, use eigenvects. eigenvects returns a list of tuples of the form (eigenvalue, algebraic_multiplicity, [eigenvectors]).
M.eigenvects()
[(-2,
1,
[Matrix([
[0],
[1],
[1],
[1]])]),
(3,
1,
[Matrix([
[1],
[1],
[1],
[1]])]),
(5,
2,
[Matrix([
[1],
[1],
[1],
[0]]),
Matrix([
[ 0],
[-1],
[ 0],
[ 1]])])]
This shows us that, for example, the eigenvalue 5 also has geometric multiplicity 2, because it has two eigenvectors. Because the algebraic and geometric multiplicities are the same for all the eigenvalues, M is diagonalizable.
To diagonalize a matrix, use diagonalize. diagonalize returns a tuple \((P,D)\), where \(D\) is diagonal and \(M=PDP^{−1}\).
P, D = M.diagonalize()
P
D
P*D*P**-1
P*D*P**-1 == M
True
Note that since eigenvects also includes the eigenvalues, you should use it instead of eigenvals if you also want the eigenvectors. However, as computing the eigenvectors may often be costly, eigenvals should be preferred if you only wish to find the eigenvalues.
If all you want is the characteristic polynomial, use charpoly. This is more efficient than eigenvals, because sometimes symbolic roots can be expensive to calculate.
lamda = symbols('lamda')
p = M.charpoly(lamda)
factor(p.as_expr())