|
Finds the proper values and proper vectors
("eigenvalues" and "eigenvectors") of a given square matrix, A,
by the method of A. M. Danilevsky [Faddeeva, 1959]. See a sketch of the
theory.pdf.
The basis data are the "Le Verrier's matrix",
with characteristic polynomial coefficients,
after 1: (47.888, 797.279, 5349.455, 12296.55).
1) Matrices with identical proper values
are not solved here. A perturbation of the matrix is suggested.
2) If there is no alternative for a pivot of 0 (to its left),
the procedure stops, but leaves a "split" problem
(as in the last example below).
Suggested alternative examples, with results
(set of proper values, λ, and proper vectors, E-vectors):
| Matrix | λ |
E-vectors |
2 1
1 2 | 3
1 |
1 1
1 -1 | | |
| Matrix | λ |
E-vectors |
1 2
2 1 | 3
-1 |
1 1
1 -1 | | |
| Matrix | λ |
E-vectors |
2 -4
-1 -1 | 3
-2 |
1 1
-1⁄4 1 | | |
| Matrix | λ |
E-vectors |
5 8 16
4 1 8
-4 -4 -11 |
1
-3
-3 |
1 -1 1
1⁄2 1 0
-1⁄2 0 -1⁄2 |
|
[Or the last example with last row -4 0 11, Λ = (5 -3 -7).]
Another, order 4 Excel example solution.xls...
* We adopted the name in the version
in Faddeeva [1959]. |
• Faddeeva,
V. N., 1959,
"Computational methods of Linear Algebra", Dover Publications, Inc.,
New York, NY (USA),
pp 166–176.pdf.
ISBN: 0-486-60424-1.
• Danilevskii
(Данилевский),
A. M., 1937, "The numerical solution of the secular equation" (Russian),
Matem. Sbornik, 44(2), pp 169–171.
• Cited by Faddeeva [1959]:
Krylov, Aleksei Nikolaevich (1863–1945);
Le Verrier, Urbain Jean Joseph
(1811–1877).
• Frobenius, Ferdinand Georg (1849-10-26, 1917-08-03);
Wikipedia.
Search Frobenius form...
• Carter, Tamara A.,
Richard A. Tapia,
Anne Papakonstantinou,
"Linear Algebra: an introduction to Linear Algebra for pre-calculus students",
Rice University.
• Malek,
Massoud (with sound),
"Characteristic Polynomial".pdf
(=) ←
California State University, East Bay.
• 1893-02-03: Duhamel, Jean Marie Constant
(1797-02-05). |
