A First Course in Linear Algebra, 2017a

358 Spectral Theory
Notice that the above equation can be rearranged as A = PDP −1 . Suppose we wanted to compute
A 100 . By diagonalizing A first it suffices to then compute ( PDP −1) 100 , which reduces to PD 100 P −1 .This
last computation is much simpler than A 100 . While this process is described in detail later, it provides
motivation for diagonalization.
7.2.2 Diagonalizing a Matrix
The most important theorem about diagonalizability is the following major result.
Theorem 7.18: Eigenvectors and Diagonalizable Matrices
An n × n matrix A is diagonalizable if and only if there is an invertible matrix P given by
P = [ X 1 X 2 ··· X n
]
where the X k are eigenvectors of A.
Moreover if A is diagonalizable, the corresponding eigenvalues of A are the diagonal entries of the
diagonal matrix D.
Proof. Suppose P is given as above as an invertible matrix whose columns are eigenvectors of A. Then
P −1 is of the form
⎡ ⎤
W1
T P −1 W T = ⎢ ⎥
⎣
2. ⎦
where Wk T X j = δ kj , which is the Kronecker’s symbol defined by
{ 1ifi = j
δ ij =
0ifi ≠ j
W T n
Then
P −1 AP =
=
=
⎡
⎢
⎣
⎡
⎢
⎣
⎡
⎢
⎣
W T 1
W T 2
.
W T n
W T 1
W T 2
.
W T n
⎤
[ ]
⎥ AX1 AX 2 ··· AX n
⎦
⎤
[ ]
⎥ λ1 X 1 λ 2 X 2 ··· λ n X n
⎦
⎤
λ 1 0
. ..
⎥
⎦
0 λ n