characteristic polynomial
Characteristic Polynomial of a Matrix
Let be a matrix over some field . The characteristic polynomial of in an indeterminate is defined by the determinant:
Remarks
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The polynomial is an th-degree polynomial over .
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The characteristic equation of is the equation , and the solutions to which are the eigenvalues of .
Characteristic Polynomial of a Linear Operator
Now, let be a linear operator on a vector space of dimension . Let and be any two ordered bases for . Then we may form the matrices and . The two matrix representations of are similar matrices, related by a change of bases matrix. Therefore, by the second remark above, we define the characteristic polynomial of , denoted by , in the indeterminate , by
The characteristic equation of is defined accordingly.
Title | characteristic polynomial |
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Canonical name | CharacteristicPolynomial |
Date of creation | 2013-03-22 12:17:47 |
Last modified on | 2013-03-22 12:17:47 |
Owner | CWoo (3771) |
Last modified by | CWoo (3771) |
Numerical id | 10 |
Author | CWoo (3771) |
Entry type | Definition |
Classification | msc 15A18 |
Related topic | Equation |
Defines | characteristic equation |