[英]The roots of the characteristic polynomial and the eigenvalues are not the same
This is matrix B
这是矩阵B
B = [1 2 0 ; 2 4 6 ; 0 6 5]
The result of eig(B)
is: eig(B)
的结果是:
{-2.2240, 1.5109, 10.7131}
and the characteristic polynomial of B
by this link is 并且该链路的B
的特征多项式是
syms x
polyB = charpoly(B,x)
x^3 - 10*x^2 - 11*x + 36
but the answer of solve(polyB)
is 但solve(polyB)
的答案是
133/(9*(3^(1/2)*5492^(1/2)*(i/3) + 1009/27)^(1/3)) + ((3^(1/2)*5492^(1/2)*i)/3 + 1009/27)^(1/3) + 10/3
(3^(1/2)*(133/(9*(3^(1/2)*5492^(1/2)*(i/3) + 1009/27)^(1/3)) - ((3^(1/2)*5492^(1/2)*i)/3 + 1009/27)^(1/3))*i)/2 - 133/(18*(3^(1/2)*5492^(1/2)*(i/3) + 1009/27)^(1/3)) - ((3^(1/2)*5492^(1/2)*i)/3 + 1009/27)^(1/3)/2 + 10/3
10/3 - 133/(18*(3^(1/2)*5492^(1/2)*(i/3) + 1009/27)^(1/3)) - ((3^(1/2)*5492^(1/2)*i)/3 + 1009/27)^(1/3)/2 - (3^(1/2)*(133/(9*(3^(1/2)*5492^(1/2)*(i/3) + 1009/27)^(1/3)) - ((3^(1/2)*5492^(1/2)*i)/3 + 1009/27)^(1/3))*i)/2
which I don't know what it is while I expect it to be the eigenvalues of B
. 我不知道它是什么,而我期望它是B
的特征值。 What is the problem? 问题是什么?
I do not understand why you add x
and symbolic maths, they are not required for your task. 我不明白为什么你要添加x
和符号数学,它们不是你的任务所必需的。
B = [1 2 0 ; 2 4 6 ; 0 6 5]
cp=charpoly(B)
eig2=roots(cp)
returns: 收益:
eig2 =
10.7131
-2.2240
1.5109
However, if for some reason you insist in using symbolic (which you should not for a numerical task), you can do 但是,如果由于某种原因你坚持使用符号(你不应该用于数字任务),你可以这样做
double(solve(polyB))
ans =
10.7131 + 0.0000i
-2.2240 - 0.0000i
1.5109 - 0.0000i
(note imaginary parts is zero) (注意虚部为零)
Since I do not have MATLAB in this machine, I will use SymPy instead: 由于我在这台机器上没有MATLAB,我将使用SymPy代替:
>>> from sympy import *
>>> B = Matrix([[1, 2, 0],
[2, 4, 6],
[0, 6, 5]])
Computing the characteristic polynomial and its roots: 计算特征多项式及其根:
>>> s = Symbol('s')
>>> p = (s*eye(3) - B).det()
>>> p
s**3 - 10*s**2 - 11*s + 36
>>> roots = solve(p,s)
Computing floating-point approximations of the three roots: 计算三个根的浮点近似值:
>>> [ r.evalf() for r in roots ]
[1.51092975992931 - 0.e-22*I, -2.22404024437578 + 0.e-22*I, 10.7131104844465 - 0.e-20*I]
Since B
is symmetric, its eigenvalues must be real. 由于B
是对称的,因此其特征值必须是实数。 Note that the imaginary parts of the floating-point approximations of the roots are indeed equal to zero. 注意,根的浮点近似的虚部确实等于零。
Printing in LaTeX, the exact values of the roots are: 在LaTeX中打印,根的确切值是:
Note that some roots are "longer" than others, ie, they require more symbols. 请注意,某些根比其他根“更长”,即它们需要更多符号。 However, they are exact . 但是,它们是准确的 。 Using floating-point arithmetic, all roots have the same "size", but they are approximations. 使用浮点运算,所有根都具有相同的“大小”,但它们是近似值。
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