Similarity: diagonalizability, eigenvalues and

eigenvectors.

Birzhan Kalmurzayev

Ac. year 2023-2024

Birzhan Kalmurzayev
Similarity: diagonalizability, eigenvalues and eigenvectors.
 We’ve defined two matrices H and Ĥ to be matrix equivalent if
there are non-singular P and Q such that Ĥ = PHQ. We were
motivated by this diagram showing both H and Ĥ representing a
map, h but with respect to different pairs of bases, B, D and B̂, D̂.

Birzhan Kalmurzayev
Similarity: diagonalizability, eigenvalues and eigenvectors.
 We now consider the special case where the codomain equals the
domain and in particular we add the requirement that the
codomain’s basis equals the domain’s basis, so we are considering
representations with respect to B, B and D, D.

Birzhan Kalmurzayev
Similarity: diagonalizability, eigenvalues and eigenvectors.
 We now consider the special case where the codomain equals the
domain and in particular we add the requirement that the
codomain’s basis equals the domain’s basis, so we are considering
representations with respect to B, B and D, D.

In matrix terms,

RepD,D (t) = RepB,D (id) · RepB,B (t) · (RepB,D (id))−1 .

Birzhan Kalmurzayev
Similarity: diagonalizability, eigenvalues and eigenvectors.
Similar matrices

Definition
The matrices T and S are similar if there is a non-singular P such
that T = P · S · P −1

Birzhan Kalmurzayev
Similarity: diagonalizability, eigenvalues and eigenvectors.
Similar matrices

Definition
The matrices T and S are similar if there is a non-singular P such
that T = P · S · P −1
Example. Assume
   
2 1 2 −3
P= , S= .
1 1 1 −1

Find a matrix T similar to S with the matrix P.

Birzhan Kalmurzayev
Similarity: diagonalizability, eigenvalues and eigenvectors.
Properties

Theorem
The relation ”Similar to” is an equivalence relation.

Example
The only matrix similar to the zero matrix is itself:
PZP −1 = Z ;
The only matrix similar to the identity matrix is itself:
PIP −1 = I.

Corollary
Relations ”matrix equivalent” and ”similar” are different.

Birzhan Kalmurzayev
Similarity: diagonalizability, eigenvalues and eigenvectors.
Diagonalizability

Definition
A transformation is diagonalizable if it has a diagonal
representation with respect to the same basis for the codomain as
for the domain. A diagonalizable matrix is one that is similar to a
diagonal matrix: T is diagonalizable if there is a non-singular P
such that PTP −1 is diagonal.

Birzhan Kalmurzayev
Similarity: diagonalizability, eigenvalues and eigenvectors.
Diagonalizability

Definition
A transformation is diagonalizable if it has a diagonal
representation with respect to the same basis for the codomain as
for the domain. A diagonalizable matrix is one that is similar to a
diagonal matrix: T is diagonalizable if there is a non-singular P
such that PTP −1 is diagonal.

Example: The following matrix is diagonalizable

 
4 −2
1 1

Birzhan Kalmurzayev
Similarity: diagonalizability, eigenvalues and eigenvectors.
Diagonalizability

Definition
A transformation is diagonalizable if it has a diagonal
representation with respect to the same basis for the codomain as
for the domain. A diagonalizable matrix is one that is similar to a
diagonal matrix: T is diagonalizable if there is a non-singular P
such that PTP −1 is diagonal.

Example: The following matrix is diagonalizable

 
4 −2
1 1
 
−1 2
Hint: Consider P =
1 −1
Birzhan Kalmurzayev
Similarity: diagonalizability, eigenvalues and eigenvectors.
 Lemma
A transformation t is diagonalizable if and only if there is a basis
B = ⟨β1 , β2 , . . . , βn ⟩ and scalars λ1 , λ2 , . . . , λn such that
t(βi ) = λi · βi for each i.

Birzhan Kalmurzayev
Similarity: diagonalizability, eigenvalues and eigenvectors.
 Lemma
A transformation t is diagonalizable if and only if there is a basis
B = ⟨β1 , β2 , . . . , βn ⟩ and scalars λ1 , λ2 , . . . , λn such that
t(βi ) = λi · βi for each i.

Example. Diagonalize the matrix

 
3 2
T =
0 1

Birzhan Kalmurzayev
Similarity: diagonalizability, eigenvalues and eigenvectors.
Eigenvalues and Eigenvectors

Definition
A transformation t : V → V has a scalar eigenvalue λ if there is a
non-zero eigenvector ξ ∈ V such that t(ξ) = λ · ξ.

Definition
A square matrix T has a scalar eigenvalue associated with the
non-zero eigenvector if T · ξ = λ · ξ.

Birzhan Kalmurzayev
Similarity: diagonalizability, eigenvalues and eigenvectors.
Example 1.

Assume  
1 2 1
T =  2 0 −2
−1 2 3
Find eigenvalues and eigenvectors associated the eigenvalues.

Birzhan Kalmurzayev
Similarity: diagonalizability, eigenvalues and eigenvectors.
Characteristic polynomial

Definition
The characteristic polynomial of a square matrix T is the
determinant |T − xI| where x is a variable. The characteristic
equation is |T − xI| = 0. The characteristic polynomial of a
transformation t is the characteristic polynomial of any matrix
representatation RepB,B (t).

Lemma
A linear transformation on a nontrivial vector space has at least
one eigenvalue.

Birzhan Kalmurzayev
Similarity: diagonalizability, eigenvalues and eigenvectors.
Eigenspace

Definition
The eigenspace of a transformation t associated with the
eigenvalue λ is Vλ = {ξ : t(ξ) = λξ}. The eigenspace of a matrix
is analogous.

Lemma
An eigenspace is a subspace.

Birzhan Kalmurzayev
Similarity: diagonalizability, eigenvalues and eigenvectors.
Eigenspace

Definition
The eigenspace of a transformation t associated with the
eigenvalue λ is Vλ = {ξ : t(ξ) = λξ}. The eigenspace of a matrix
is analogous.

Lemma
An eigenspace is a subspace.

Example: In Example 1 find eigenspaces associated with the

eigenvalues 0 and 2.

Birzhan Kalmurzayev
Similarity: diagonalizability, eigenvalues and eigenvectors.
 Theorem
For any set of distinct eigenvalues of a map or matrix, a set of
associated eigenvectors, one per eigenvalue, is linearly independent.

Corollary
An n × n matrix with n distinct eigenvalues is diagonalizable.

Birzhan Kalmurzayev
Similarity: diagonalizability, eigenvalues and eigenvectors.
 Theorem
For any set of distinct eigenvalues of a map or matrix, a set of
associated eigenvectors, one per eigenvalue, is linearly independent.

Corollary
An n × n matrix with n distinct eigenvalues is diagonalizable.

Example: For the matrix

 
2 −2 2
T = 0 1 1
−4 8 3

Find all eigenvalues and bases for eigenspaces associated the

eigenvalues. Diagonalize if it possible.

Birzhan Kalmurzayev
Similarity: diagonalizability, eigenvalues and eigenvectors.