Diagonalizability
3 important questions on Diagonalizability
When is T diagonalizable?
If T is a linear operator over n dimensional v.s. , then T is diagonalizable if it has n distinct eigenvalues.
-Suppose T has n distict eigenvalues we know {v1,v2,...,vn} is linearly independentand since dim(V)=n, this set is a basis for V so T is diagonalizable.
-Suppose T has n distict eigenvalues we know {v1,v2,...,vn} is linearly independentand since dim(V)=n, this set is a basis for V so T is diagonalizable.
What that does it mean for a polynomial to split over?
-A polynomial splits over F if there exists scalars c,aq,a2...,an in F such that
f(t)= c(t-a1)(t-a2)...(t-an)
- A characteristic polynomial of any diagonalizable linear operator on a vector space V over a field F splits over F.
f(t)= c(t-a1)(t-a2)...(t-an)
- A characteristic polynomial of any diagonalizable linear operator on a vector space V over a field F splits over F.
What are some properties of T being diagonalizable?
-T is diagonalizable iff the multiplicity of Ri = dim(Eri) for all i.
- If T is diagonalizable and Bi is an ordered basis for Eri for each i, then B= B1 U B2 U ... U Bk is an orderes basis for V consisting of eigenvectors of T
- If T is diagonalizable and Bi is an ordered basis for Eri for each i, then B= B1 U B2 U ... U Bk is an orderes basis for V consisting of eigenvectors of T
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