Page 284 - A Course in Linear Algebra with Applications

268 Chapter Eight: Eigenvectors and Eigenvalues

should give an example to demonstrate this. It is an important
observation that if A is diagonalizable and its eigenvalues are
c\,..., c n , then A must be similar to the diagonal matrix with
on the principal diagonal. This is because similar
ci,..., c n
matrices have the same eigenvalues and the eigenvalues of a
diagonal matrix are just the entries on the principal diagonal
- see Example 8.1.2.
What we are aiming for is a criterion which will tell us
exactly which matrices are diagonalizable. A key step in the
search for this criterion comes next.

Theorem 8.1.5
Let A be an n x n matrix over a field F and let C\,..., c r
be distinct eigenvalues of A with associated eigenvectors
Xi,..., X r. Then {Xi,..., X r} is a linearly independent sub-
n
set of F .
Proof
Assume the theorem is false; then there is a positive integer
i such that {X±,... ,Xi} is linearly independent, but the ad-
dition of the next vector X i+i produces a linearly dependent
set {Xi,..., Xi+x}. So there are scalars d\,..., rfj+i, not all
of them zero, such that

hXi + • • • + di +1X i+1 = 0 .

Premultiply both sides of this equation by A and use the equa-
tions AXj = CjXj to get

CidiXi -I 1- c i+1d i+1X i+i — 0.

On subtracting Q+I times the first equation from the second,
we arrive at the relation

(ci - c i+1 )diXi H h (CJ - c i+i)diXi = 0.