An Intuitive Derivation of Eigenvectors
Earlier, the matrix F for the function f was represented by:
When I use the basis B={[10],[11]}, the matrix FB in basis B becomes:
More generally, for a basis B={b1,b2}, the matrix is:
We took this short detour into notation for a very specific reason – rewriting a matrix in a different basis is actually a neat trick that allows us to reconfigure the matrix to make it easier to use. We saw earlier that choosing a new basis B={b1,b2} creates a new coordinate axis for R2 like below:
In the graph above, we can see that [λ10]B=λ1b1, so
From the above, we see clearly that [0λ2]B=λ2b2, so
So if we can find a basis B formed by b1 and b2 such that:
Is there a special name for the vectors above b1 and b2 that magically let us rewrite a matrix as a diagonal? We need to rewrite these vectors in the notation for our new basis B.
Putting this all together, FB=[f(b1)Bf(b2)B] FB=[1003] So we get the nice diagonal we wanted!
Source: dhruvp.netlify.com