Revision history [back]
 | 1 | initial version |
Hello, can you provide a link to the page where you saw eigenvalues of a symbolic matrix calculated?
What you could always compute is the characteristic polynomial, but unless the dimension is very slow, how would you express the eigenvalues?
Below a computation of the characteristic polynomial. I use a polynomial ring instead of the symbolic ring, this is usually a better idea.
sage: R.<s,b> = QQ[]
sage: M = MatrixSpace(R,10)
sage: A = M([[0,0,b,b,b,b,b,b,b,b],
....: [s,0,0,0,0,0,0,0,0,0],
....: [0,s,0,0,0,0,0,0,0,0],
....: [0,0,s,0,0,0,0,0,0,0],
....: [0,0,0,s,0,0,0,0,0,0],
....: [0,0,0,0,s,0,0,0,0,0],
....: [0,0,0,0,0,s,0,0,0,0],
....: [0,0,0,0,0,0,s,0,0,0],
....: [0,0,0,0,0,0,0,s,0,0],
....: [0,0,0,0,0,0,0,0,s,0]])
sage: A
[0 0 b b b b b b b b]
[s 0 0 0 0 0 0 0 0 0]
[0 s 0 0 0 0 0 0 0 0]
[0 0 s 0 0 0 0 0 0 0]
[0 0 0 s 0 0 0 0 0 0]
[0 0 0 0 s 0 0 0 0 0]
[0 0 0 0 0 s 0 0 0 0]
[0 0 0 0 0 0 s 0 0 0]
[0 0 0 0 0 0 0 s 0 0]
[0 0 0 0 0 0 0 0 s 0]
sage: A.charpoly()
x^10 - s^2*b*x^7 - s^3*b*x^6 - s^4*b*x^5 - s^5*b*x^4 - s^6*b*x^3 - s^7*b*x^2 - s^8*b*x - s^9*b
| | 2 | No.2 Revision |
Hello, can you provide a link to the page where you saw eigenvalues of a symbolic matrix calculated?
What you could always compute is the characteristic polynomial, but unless the dimension is very slow, low, how would you express the eigenvalues?
Below a computation of the characteristic polynomial. I use a polynomial ring instead of the symbolic ring, this is usually a better idea.
sage: R.<s,b> = QQ[]
sage: M = MatrixSpace(R,10)
sage: A = M([[0,0,b,b,b,b,b,b,b,b],
....: [s,0,0,0,0,0,0,0,0,0],
....: [0,s,0,0,0,0,0,0,0,0],
....: [0,0,s,0,0,0,0,0,0,0],
....: [0,0,0,s,0,0,0,0,0,0],
....: [0,0,0,0,s,0,0,0,0,0],
....: [0,0,0,0,0,s,0,0,0,0],
....: [0,0,0,0,0,0,s,0,0,0],
....: [0,0,0,0,0,0,0,s,0,0],
....: [0,0,0,0,0,0,0,0,s,0]])
sage: A
[0 0 b b b b b b b b]
[s 0 0 0 0 0 0 0 0 0]
[0 s 0 0 0 0 0 0 0 0]
[0 0 s 0 0 0 0 0 0 0]
[0 0 0 s 0 0 0 0 0 0]
[0 0 0 0 s 0 0 0 0 0]
[0 0 0 0 0 s 0 0 0 0]
[0 0 0 0 0 0 s 0 0 0]
[0 0 0 0 0 0 0 s 0 0]
[0 0 0 0 0 0 0 0 s 0]
sage: A.charpoly()
x^10 - s^2*b*x^7 - s^3*b*x^6 - s^4*b*x^5 - s^5*b*x^4 - s^6*b*x^3 - s^7*b*x^2 - s^8*b*x - s^9*b
| | 3 | No.3 Revision |
Hello, can you provide a link to the page where you saw eigenvalues of a symbolic matrix calculated?
What you could always compute is the characteristic polynomial, but unless the dimension is very low, how would you express the eigenvalues?
Below a computation of the characteristic polynomial. I use a polynomial ring instead of the symbolic ring, this is usually a better idea.
sage: R.<s,b> = QQ[]
sage: M = MatrixSpace(R,10)
sage: A = M([[0,0,b,b,b,b,b,b,b,b],
....: [s,0,0,0,0,0,0,0,0,0],
....: [0,s,0,0,0,0,0,0,0,0],
....: [0,0,s,0,0,0,0,0,0,0],
....: [0,0,0,s,0,0,0,0,0,0],
....: [0,0,0,0,s,0,0,0,0,0],
....: [0,0,0,0,0,s,0,0,0,0],
....: [0,0,0,0,0,0,s,0,0,0],
....: [0,0,0,0,0,0,0,s,0,0],
....: [0,0,0,0,0,0,0,0,s,0]])
sage: A
[0 0 b b b b b b b b]
[s 0 0 0 0 0 0 0 0 0]
[0 s 0 0 0 0 0 0 0 0]
[0 0 s 0 0 0 0 0 0 0]
[0 0 0 s 0 0 0 0 0 0]
[0 0 0 0 s 0 0 0 0 0]
[0 0 0 0 0 s 0 0 0 0]
[0 0 0 0 0 0 s 0 0 0]
[0 0 0 0 0 0 0 s 0 0]
[0 0 0 0 0 0 0 0 s 0]
sage: A.charpoly()
x^10 - s^2*b*x^7 - s^3*b*x^6 - s^4*b*x^5 - s^5*b*x^4 - s^6*b*x^3 - s^7*b*x^2 - s^8*b*x - s^9*b
[Edited 2014-12-13 to take comments into account.]
The eigenvalues are the roots of the characteristic polynomial. As @Dima points out, for polynomials of degree up to three there are explicit formulas for finding the roots. In higher degree, some polynomials of very special types still will let you extract their roots with formulas, but most won't.
Here, the polynomial has a fairly simple form and is inviting us to play further.
