Page 415 - 35Linear Algebra

G.12 Diagonalization 415

G.12 Diagonalization

Non Diagonalizable Example

First recall that the derivative operator is linear and that we can write it
as the matrix
 
0 1 0 0 · · ·
 0 0 2 0
d · · · 
 
=  0 0 0 3 · · ·  .
dx  
. . . . .
. . . . .
. . . . .
We note that this transforms into an infinite Jordan cell with eigenvalue 0
or
 
0 1 0 0 · · ·
 0 0 1 0 · · · 
 
 0 0 0 1 · · · 
 
. . . . .
. . . . .
. . . . .
which is in the basis {n −1 n
x } n (where for n = 0, we just have 1). Therefore
we note that 1 (constant polynomials) is the only eigenvector with eigenvalue
0 for polynomials since they have finite degree, and so the derivative is
not diagonalizable. Note that we are ignoring infinite cases for simplicity,
but if you want to consider infinite terms such as convergent series or all
formal power series where there is no conditions on convergence, there are
many eigenvectors. Can you find some? This is an example of how things can
change in infinite dimensional spaces.
C
For a more finite example, consider the space P of complex polynomials of
3
degree at most 3, and recall that the derivative D can be written as
 
0 1 0 0
 0 0 2 0 
D =   .
 0 0 0 3 
0 0 0 0

You can easily check that the only eigenvector is 1 with eigenvalue 0 since D
always lowers the degree of a polynomial by 1 each time it is applied. Note
4
that this is a nilpotent matrix since D = 0, but the only nilpotent matrix
that is ‘‘diagonalizable’’ is the 0 matrix.

Change of Basis Example

This video returns to the example of a barrel filled with fruit

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