Difference between revisions of "Diagonalizable"
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Given this topology, the set of diagonalizeable maps is a dense subset of <math>M_n(\mathbb{C}^n)</math>. | Given this topology, the set of diagonalizeable maps is a dense subset of <math>M_n(\mathbb{C}^n)</math>. | ||
| − | We can prove this as follows: Every complex matrix <math>A</math> is conjugate to a matrix <math>B</math> in Jordan-canonical form. One can then perturb the diagonal elements <math> | + | We can prove this as follows: Every complex matrix <math>A</math> is conjugate to a matrix <math>B</math> in Jordan-canonical form. One can then perturb the diagonal elements <math>b_{ii}</math> of <math>B</math> by arbitrarily small numbers <math>\epsilon_i</math> so that the diagonal elements <math>b_{ii}+\epsilon_i</math>of the perturbed matrix are distinct. But this implies that the perturbed matrix is diagonalizeable. Thus, we can find a diagonalizeable matrix arbitrarily close to a conjugate of <math>A</math>. But since conjugation is a length-preserving operation on the inner product space of complex matrices, this shows that <math>A</math> is arbitrarily close to a diagonalizeable matrix. This completes the proof. |
[[Category:Mathematics]] | [[Category:Mathematics]] | ||
Revision as of 02:13, July 5, 2008
An operator T on a finite-dimensional vectorspace V is diagonalizeable if V has a basis of eigenvectors for T.
Denseness of diagonalizeable operators
The space of complex linear operators on 
may be identified with the vectorspace 
of nxn matrices with complex coefficients. As such, it inherits a natural structure as a topological space.
Given this topology, the set of diagonalizeable maps is a dense subset of 
.
We can prove this as follows: Every complex matrix 
is conjugate to a matrix 
in Jordan-canonical form. One can then perturb the diagonal elements 
of by arbitrarily small numbers 
so that the diagonal elements 
of the perturbed matrix are distinct. But this implies that the perturbed matrix is diagonalizeable. Thus, we can find a diagonalizeable matrix arbitrarily close to a conjugate of . But since conjugation is a length-preserving operation on the inner product space of complex matrices, this shows that is arbitrarily close to a diagonalizeable matrix. This completes the proof.
