6.11.3. Theorem 6.10-12.
Definition
Two n  n matrices A and B are similar if there exists a nonsingular matrix C such that
B  C 1 AC .
Theorem 6.10.
Two n  n matrices are similar iff they represent the same linear transformation.
Proof
This is just the combination of Theorems 6.8-9.
Comment
Similar matrices share many properties.
since, by Theorems 5.11-12, we have,
For example, they have the same determinant
det  C 1 AC   det  C 1   det A det C   det A
(a)
Theorem 6.11.
Similar matrices have the same characteristic polynomial and hence the same
eigenvalues.
Proof
Let A and B be similar so that B  C 1 AC . Then
 I  B   I  C 1 AC  C 1   I  A C
The theorem is thus proved with the help of eq(a).
Comment
Theorems 6.10 and 6.11 show that all matrix representations of a given linear
transformation T have the same characteristic polynomial f    . Hence, f    is
also called the characteristic polynomial of T.
Theorem 6.12.
Let W be an n-D linear space over the scalar field F and T : W  W is a linear
transformation. If the characteristic polynomial of T has n distinct roots 1 , , n in F,
then
(a) The corresponding eigenvectors u1 ,
(b) m T     diag  1,
, un form a basis for W.
, n  relative to the ordered basis U   u1,
(c) If m T   A relative to another ordered basis V   v1,
, un  .
, vn  , then   C 1 AC ,
where C is the transition matrix given by U  VC .
Proof of (a)
By Theorem 6.7, every root i is an eigenvalue.
Since i are distinct, the
corresponding eigenvectors are by Theorem 6.2 independent.
they form a basis for W.
Since there are n of them,
Proof of (b)
Since ui is the eigenvector belonging to i, we have T  ui   i ui . Proof is completed
by comparing with the definition T  ui    uk tki , where m T    tij  .
n
k 1
Proof of (c)
Proof is furnished by Theorem 6.8.
Comment
The transition matrix C in Theorem 6.12(c) is sometimes called the diagonalizing matrix.
If V   I1,
, I n  , then the equation U  VC shows that the kth column of C consists of
the components of the eigenvector ui relative to the unit coordinate basis I1 ,
, In .
If the eigenvalues of A are all distinct, then A is similar to a diagonal matrix, i.e., it is
diagonalizable. If the eigenvalues are not distinct, then A can be diagonalized only if the
number of independent eigenvectors belonging to each eigenvalue is always equal to the
multiplicity of that eigenvalue.