Canonical Forms

Michael Freeze
MAT 531: Linear Algebra
UNC Wilmington

Spring 2015

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Canonical Forms

I Diagonal Form
I Triangular Form
I Jordan Canonical Form
I Rational Canonical Form

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Diagonal Form

Theorem
Let T be a linear operator on a finite-dimensional vector space
V such that the characteristic polynomial of T splits. If the
algebraic and geometric multiplicity of each eigenvalue of T
coincide, then there exists a basis of V for which T has a
diagonal matrix representation.

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Triangular Form

Theorem
Let T be a linear operator on a finite-dimensional vector space
V such that the characteristic polynomial of T splits. Then
there exists a basis of V for which T has a triangular matrix
representation.

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Invariant Direct-Sum Decompositions
Definition
A vector space V is said to be the direct sum of subspaces
W1 , . . . , Wr when every vector ~v ∈ V can be written uniquely
in the form ~v = w ~1 + w~2 + · · · + w ~ i ∈ Wi for
~ r with w
1 ≤ i ≤ r.
We write V = W1 ⊕ W2 ⊕ · · · ⊕ Wr .
Definition
Let V = W1 ⊕ · · · ⊕ Wr where T (Wi ) ⊆ Wi for 1 ≤ i ≤ r .
We say that T is the direct sum of the Ti when Wi is
T -invariant for 1 ≤ i ≤ r .
Taking Ti = T |Wi to be the restriction of T to Wi for
1 ≤ i ≤ r , we write T = T1 ⊕ · · · ⊕ Tr .

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Minimal Polynomial

Definition
Let T be a linear operator on a finite-dimensional vector space
V . A polynomial m(t) is called the minimal polynomial of T
when m(t) is a monic polynomial of least positive degree for
which m(T ) = T0 .

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Computation of Minimal Polynomial

Given an n × n matrix A where n is small, we compute powers

of A to obtain I, A, A2 , . . . , An . Form matrix B so that its rows
are the vectorized powers of A. Row reduce the augmented
matrix [B | In+1 ] and identify a dependency among the first p
rows of B where p is minimal.

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Example
 
1 0 3
For A =  2 1 1, obtain
−1 0 3
 
1 0 0 0 1 0 0 0 1 1 0 0 0
 1 0 3 2 1 1 −1 0 3 0 1 0 0 
[B | I4 ] = 
 −2
.
0 12 3 1 10 −4 0 6 0 0 1 0 
−14 0 30 −5 1 40 −10 0 6 0 0 0 1

From the reduced row echelon form of [B | I4 ], we obtain

g (t) = 1 − 35 t + 65 t 2 − 16 t 3 as an annihilating polynomial for A.
We obtain m(t) = t 3 − 5t 2 + 10t − 6 is the minimal
polynomial of A.

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Primary Decomposition

Theorem
Let T be a linear operator on a finite-dimensional vector space
V with minimal polynomial m(t) = f1 (t)n1 f2 (t)n2 · · · fr (t)nr
where the fi (t) are distinct monic irreducible polynomials.
Then V is the direct sum of the T -invariant subspaces
W1 , . . . , Wr where Wi is the null space of fi (T )ni for
1 ≤ i ≤ r.
Moreover, fi (t)ni is the minimal polynomial of the restriction
of T to Wi .

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Nilpotent Operators

Definition
A linear operator T on a finite-dimensional vector space V is
said to be nilpotent when T n = T0 for some positive integer n.
When for some positive integer k we have T k = T0 but
T k−1 6= T0 , we say that k is the index of nilpotency of T .

Definition
A matrix N having 1’s on the superdiagonal and 0’s elsewhere
is said to be a Jordan nilpotent block.
A matrix J(λ) having λ’s on the diagonal, 1’s on the
superdiagonal, and 0’s elsewhere is said to be a Jordan block
belonging to the eigenvalue λ.

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Jordan Canonical Form
Theorem
Let T be a linear operator on a finite-dimensional vector space
V whose characteristic and minimal polynomials are,
respectively,

f (t) = (t−λ1 )n1 · · · (t−λr )nr and m(t) = (t−λ1 )m1 · · · (t−λr )mr

where λ1 , . . . , λr are distinct scalars. Then T has a block

diagonal matrix representation in which each diagonal entry is
a Jordan block.

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Jordan Canonical Form

f (t) = (t−λ1 )n1 · · · (t−λr )nr and m(t) = (t−λ1 )m1 · · · (t−λr )mr
For each λi , the Jordan blocks belonging to λi have the
following properties:
(i) There is at least one Jordan block belonging to λi of
order mi and all other Jordan blocks belonging to λi are
of order not exceeding mi .
(ii) The sum of the orders of the Jordan blocks belonging to
λi is ni .
(iii) The number of Jordan blocks belonging to λi equals the
geometric multiplicity of λi .
(iv) The number of Jordan blocks of each possible order is
uniquely determined by T .
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Example

The following matrix is in Jordan canonical form and has 3

blocks.
 
2 1 0 0 0 0
0 2 0 0 0 0
 
0 0 3 0 0 0
A= 0 0 0

 3 1 0

0 0 0 0 3 1
0 0 0 0 0 3

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Chains of Vectors
Consider an n × n Jordan block
 
λ 1 0 ··· 0 0
0 λ 1 0 0
 
0 0 λ 0 0
J =  .. .. ..  .
 
.. ..
. . . . .
 
0 0 0 λ 1
0 0 0 ··· 0 λ
Observe that
(J − λI)~e1 = ~0
(J − λI)~e2 = e~1 =⇒ (J − λI)2 e~2 = ~0
..
.
(J − λI)~ek = e~k−1 =⇒ (J − λI)k e~k = ~0
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Example
Consider  
2 1 0 0 0 0
0 2 0 0 0 0
 
0 0 3 0 0 0
A=
0
.
 0 0 3 1 0

0 0 0 0 3 1
0 0 0 0 0 3

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Example
Consider  
2 1 0 0 0 0
0 2 0 0 0 0
 
0 0 3 0 0 0
A=
0
.
 0 0 3 1 0

0 0 0 0 3 1
0 0 0 0 0 3
Multiplication by A − 2I:

(A − 2I)~e1 = ~0
(A − 2I)~e2 = e~1

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Example
Consider  
2 1 0 0 0 0
0 2 0 0 0 0
 
0 0 3 0 0 0
A= 0
.
 0 0 3 1 0

0 0 0 0 3 1
0 0 0 0 0 3
Multiplication by A − 3I:
(A − 3I)~e3 = ~0

(A − 3I)~e4 = ~0
(A − 3I)~e5 = e~4
(A − 3I)~e6 = e~5

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Finding the Jordan Canonical Form of a Matrix

Finding a Jordan basis essentially involves identifying a

number of linearly independent “chains” of vectors
{~v1 , . . . , ~vk } such that

(A − λI)~vj = ~vj−1

for 1 ≤ j ≤ k.

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Example :: One Eigenvalue

Find a Jordan canonical form for

 
0 −3 −2
A = 1 3 1 .
1 2 3

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Example :: Two Eigenvalues

Find a Jordan canonical form for

 
1 −2 0 0
1 3 1 0
A= 0
.
1 2 1
0 −1 0 1

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Example :: Two Eigenvalues

Find a Jordan canonical form for

 
1 0 3 0
1 3 0 3
A= 0
.
0 1 0
0 0 3 1

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Example :: One Eigenvalue, Two Blocks

Find a Jordan canonical form for

 
−2 1 −1 −3
−8 3 −4 −8
A=  4 −1
.
3 4
−1 0 −1 0

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What does one do when the characteristic
polynomial does not split?

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Companion Matrix

Definition
Let p(t) = t k + ak−1 t k−1 + · · · + a1 t + a0 .
The companion matrix of p(t) is the matrix
 
0 0 0 ··· 0 −a0
 1
 0 0 ··· 0 −a1 
 0
 1 0 · · · 0 −a 2

.
· · · · · · · · · · · · · · · ··· 
 
 0 0 0 · · · 0 −ak−2 
0 0 0 · · · 1 −ak−1

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Rational Canonical Form

Theorem
Let T be a linear operator on a finite-dimensional vector space
V with minimal polynomial m(t) = f1 (t)m1 f2 (t)m2 · · · fs (t)ms
where f1 (t), . . . , fs (t) are distinct monic irreducible
polynomials.
Then T has a unique block diagonal matrix representation in
which each diagonal entry is a companion matrix.

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Example

 
17 −10
Let A = .
29 −17
Find a rational canonical form for A over R.

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