MAT223: Linear Algebra I
Week 10
November 8, 2025
MAT223
November 8, 2025
1/7
Matrix Similarity
Last time we have seen that the defining matrix of a linear transformation
depends on our choice of basis. Usually, if B and C are distinct bases for
Rn , then MF ,B and MF ,C will be different matrices. However, we don’t
want to lose track of the fact that these two matrices are related to each
other.
MAT223
November 8, 2025
2/7
Matrix Similarity
Last time we have seen that the defining matrix of a linear transformation
depends on our choice of basis. Usually, if B and C are distinct bases for
Rn , then MF ,B and MF ,C will be different matrices. However, we don’t
want to lose track of the fact that these two matrices are related to each
other.
Definition (Similar matrices)
Two n × n matrices B and C are called similar if they represent the same
linear function, but in possibly different bases, i.e. there is a single linear
function F : Rn → Rn so that
MF ,B = B and MF ,C = C ,
where B and C are bases for Rn .
MAT223
November 8, 2025
2/7
An algebraic method to detect matrix similarity
Theorem
Two n × n matrices B and C are similar if and only if there exists an
invertible n × n matrix P so that B = P −1 CP.
MAT223
November 8, 2025
3/7
An algebraic method to detect matrix similarity
Theorem
Two n × n matrices B and C are similar if and only if there exists an
invertible n × n matrix P so that B = P −1 CP.
So far it is generally challenging for us to determine whether two matrices
are similar. In the rest of this week, we will develop a method to detect
matrix similarity for a special family of matrices.
MAT223
November 8, 2025
3/7
An algebraic method to detect matrix similarity
Theorem
Two n × n matrices B and C are similar if and only if there exists an
invertible n × n matrix P so that B = P −1 CP.
So far it is generally challenging for us to determine whether two matrices
are similar. In the rest of this week, we will develop a method to detect
matrix similarity for a special family of matrices.
CAP 10.4
For an n × n matrix A, if {⃗v1 , . . . , ⃗vn } is a linearly independent set of
eigenvectors, then A is similar to a matrix of the form


λ1 0 · · · 0
 0 λ2 · · · 0 


 ..
.. . .
..  .
.
. .
.
0 0 · · · λn
MAT223
November 8, 2025
3/7
Diagonal matrices
Definition (Diagonal matrices)
A matrix D is called diagonal if the
 only nonzero entries
 in the matrix
d1 0 · · · 0
 0 d2 · · · 0 


appear on the diagonal, i.e. D =  .
.. . .
.. .
 ..
.
.
.
0
MAT223
0
···
dn
November 8, 2025
4/7
Diagonal matrices
Definition (Diagonal matrices)
A matrix D is called diagonal if the
 only nonzero entries
 in the matrix
d1 0 · · · 0
 0 d2 · · · 0 


appear on the diagonal, i.e. D =  .
.. . .
.. .
 ..
.
.
.
0 0 · · · dn
In this case, we write D = diag(d1 , d2 , . . . , dn ), and call the matrix
transformation TD a dilation transformation.
MAT223
November 8, 2025
4/7
Diagonal matrices
Definition (Diagonal matrices)
A matrix D is called diagonal if the
 only nonzero entries
 in the matrix
d1 0 · · · 0
 0 d2 · · · 0 


appear on the diagonal, i.e. D =  .
.. . .
.. .
 ..
.
.
.
0 0 · · · dn
In this case, we write D = diag(d1 , d2 , . . . , dn ), and call the matrix
transformation TD a dilation transformation.
Definition (Diagonalizable matrices)
An n × n matrix A is called diagonalizable if it is similar to a diagonal
matrix.
MAT223
November 8, 2025
4/7
Diagonalization theorem
The following theorem will help us characterize which matrices are
diagonalizable:
Theorem (The Diagonalization Theorem)
An n × n matrix A is diagonalizable if and only if A has n linearly
independent eigenvectors. In this case, there are linearly independent
eigenvectors ⃗v1 , . . . , ⃗vn with corresponding eigenvalues λ1 , . . . , λn for A so
that D = C −1 AC where
D = diag(λ1 , . . . , λn ) and C = ⃗v1 · · · ⃗vn .
MAT223
November 8, 2025
5/7
Diagonalization theorem
The following theorem will help us characterize which matrices are
diagonalizable:
Theorem (The Diagonalization Theorem)
An n × n matrix A is diagonalizable if and only if A has n linearly
independent eigenvectors. In this case, there are linearly independent
eigenvectors ⃗v1 , . . . , ⃗vn with corresponding eigenvalues λ1 , . . . , λn for A so
that D = C −1 AC where
D = diag(λ1 , . . . , λn ) and C = ⃗v1 · · · ⃗vn .
To use this theorem more efficiently, it would be useful to have methods to
detect when A has enough linearly independent eigenvectors.
MAT223
November 8, 2025
5/7
Enough linearly independent eigenvectors?
Proposition
Let λ1 , . . . , λk be distinct eigenvalues of a matrix A, and suppose that
⃗vi ∈ Eλi for each i ∈ {1, . . . , k}. Then {⃗v1 , . . . , ⃗vk } is a linearly
independent set.
MAT223
November 8, 2025
6/7
Enough linearly independent eigenvectors?
Proposition
Let λ1 , . . . , λk be distinct eigenvalues of a matrix A, and suppose that
⃗vi ∈ Eλi for each i ∈ {1, . . . , k}. Then {⃗v1 , . . . , ⃗vk } is a linearly
independent set.
Recall that for an eigenvalue λ of A, the geometric multiplicity of λ is the
dimension of the λ-eigenspace Eλ .
Proposition
An n × n matrix A is diagonalizable if and only if the sum of the geometric
multiplicities of A is equal to n.
MAT223
November 8, 2025
6/7
Eigendecompositions
Note that if A is diagonalizable with D = C −1 AC , we can write
A = CDC −1 . This gives a useful decomposition of our matrix A.
MAT223
November 8, 2025
7/7
Eigendecompositions
Note that if A is diagonalizable with D = C −1 AC , we can write
A = CDC −1 . This gives a useful decomposition of our matrix A.
Definition
Suppose that A is an n × n diagonalizable matrix with eigenvalues
λ1 , . . . , λn and corresponding linearly independent eigenvectors ⃗v1 , . . . , ⃗vn .
Let D = diag(λ1 , . . . , λn ) and C = ⃗v1 · · · ⃗vn . We call the equality
A = CDC −1 the eigendecomposition of the matrix A.
MAT223
November 8, 2025
7/7
Eigendecompositions
Note that if A is diagonalizable with D = C −1 AC , we can write
A = CDC −1 . This gives a useful decomposition of our matrix A.
Definition
Suppose that A is an n × n diagonalizable matrix with eigenvalues
λ1 , . . . , λn and corresponding linearly independent eigenvectors ⃗v1 , . . . , ⃗vn .
Let D = diag(λ1 , . . . , λn ) and C = ⃗v1 · · · ⃗vn . We call the equality
A = CDC −1 the eigendecomposition of the matrix A.
Note that the eigendecomposition can also help us compute large powers
of a matrix:
Proposition
Let A be a diagonalizable matrix with eigendecomposition A = CDC −1 .
Then An = CD n C −1 for any integer n.
MAT223
November 8, 2025
7/7