Question

Explain why diagonalization is the same as changing to a basis of eigenvectors, computing the linear transformation with respect to this basis, and then changing back to the standard basis. In addition, create your own example (with specific numbers) that illustrates your explanation. You should write in complete sentences, though you can certainly have equations and symbols as part of your sentences (and probably should).

Answer 1

Given That,

Diagonalization is the same as changing to a basis of eigenvectors

the linear transformation with respect to this basis, and then changing back to the standard basis

This is easier if one starts out talking about linear transformations, and only later about matrices. A linear operator T:V→VT:V→V is diagonalisable if and only if VV admits a basis of eigenvectors for TT. Expresssing TT on a basis BB results in a square matrix that is diagonal if and only if the vectors of BBare all eigenvectors (this is immadiate from the definitions). So TT is diagonalisable if and only if its matrix with respect to some basis is diagonal (and this happens precisely for those bases that consist entirely of eigenvectors for TT).

Now if the matrix of TT one some random basis is AA, then saying that TT is diagonalisable means that some bases change applied to AA (namely one to a basis of eigenvectors) must give a diagonal matrix

The formula for base change is that AA becomes P−1APP−1AP where PP is the matrix whose columns express the coordinates of the new basis vectors with resect to the old basis. Thus a matrix is diagonalisable if and only if P−1APP−1AP is diagonal for some invertible matrix P P.

Note that the adjective "diagonalisable" refers to what happen to matrices under base change, but the property it expresses is one that applies fundamentally to a linear transformations (namely having