7.3: Diagonal matrices
( \newcommand{\kernel}{\mathrm{null}\,}\)
Note that if T has n=dim(V) distinct eigenvalues, then there exists a basis (v1,…,vn) of Vsuch that
Tvj=λjvj,for all j=1,2,…,n.
Then any v∈V can be written as a linear combination v=a1v1+⋯+anvnof v1,…,vn. Applying T to this, we obtain Tv=λ1a1v1+⋯+λnanvn.
Hence the vector M(v)=[a1⋮an]
is mapped to M(Tv)=[λ1a1⋮λnan].
This means that the matrix M(T)for T with respect to the basis of eigenvectors (v1,…,vn) is diagonal, and so we call T diagonalizable: M(T)=[λ10⋱0λn].
We summarize the results of the above discussion in the following Proposition.
Proposition 7.3.1. If T∈L(V,V) has dim(V) distinct eigenvalues, then M(T) is diagonal with respect to some basis of V. Moreover, V has a basis consisting of eigenvectors of T.
Contributors
- Isaiah Lankham, Mathematics Department at UC Davis
- Bruno Nachtergaele, Mathematics Department at UC Davis
- Anne Schilling, Mathematics Department at UC Davis
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