aarjaneiro/planetmath huggingface dataset:
generalized eigenspace

generalized eigenspace

Let

V

be a

vector space

(over a field

k

), and

T

a

linear operator

on

V

, and

λ

an

eigenvalue

of

T

. The set

E

λ

of all

generalized eigenvectors

of

T

corresponding to

λ

, together with the

zero vector

0

, is called the

generalized eigenspace

of

T

corresponding to

λ

. In short, the generalized eigenspace of

T

corresponding to

λ

is the set

E

λ

:=

{

v

∈

V

∣

(

T

-

λ

⁢

I

)

i

⁢

(

v

)

=

0

⁢

for some positive integer

⁢

i

}

.

Here are some

properties

of

E

λ

:

1.

W

λ

⊆

E

λ

, where

W

λ

is the

eigenspace

of

T

corresponding to

λ

.

2.

E

λ

is a

subspace

of

V

and

E

λ

is

T

-

invariant

.

3.

If

V

is finite dimensional, then

dim

⁡

(

E

λ

)

is the

algebraic multiplicity

of

λ

.

4.

E

λ

1

∩

E

λ

2

=

0

iff

λ

1

≠

λ

2

. More generally,

E

A

∩

E

B

=

0

iff

A

and

B

are disjoint sets of eigenvalues of

T

, and

E

A

(or

E

B

) is defined as the sum of all

E

λ

, where

λ

∈

A

(or

B

).

5.

If

V

is finite dimensional and

T

is a linear operator on

V

such that its

characteristic polynomial

p

T

splits (over

k

), then

V

=

⊕

λ

∈

S

E

λ

,

where

S

is the set of all eigenvalues of

T

.

6.

Assume that

T

and

V

have the same properties as in (5). By the

Jordan canonical form theorem

, there exists an

ordered basis

β

of

V

such that

[

T

]

β

is a

Jordan canonical form

. Furthermore, if we set

β

i

=

β

∩

E

λ

i

, then

[

T

|

E

λ

i

]

β

i

, the

matrix representation

of

T

|

E

λ

, the

restriction

of

T

to

E

λ

i

, is a Jordan canonical form. In other words,

[

T

]

β

=

(

J

1

O

⋯

O

O

J

2

⋯

O

⋮

⋮

⋱

⋮

O

O

⋯

J

n

)

where each

J

i

=

[

T

|

E

λ

i

]

β

i

is a Jordan canonical form, and

O

is a

zero matrix

.

7.

Conversely, for each

E

λ

i

, there exists an ordered basis

β

i

for

E

λ

i

such that

J

i

:=

[

T

|

E

λ

i

]

β

i

is a Jordan canonical form. As a result,

β

:=

⋃

i

=

1

n

β

i

with linear order extending each

β

i

, such that

v

i

<

v

j

for

v

i

∈

β

i

and

v

j

∈

β

j

for

i

<

j

, is an ordered basis for

V

such that

[

T

]

β

is a Jordan canonical form, being the

direct sum of matrices

J

i

.

8.

Each

J

i

above can be further decomposed into

Jordan blocks

, and it turns out that the number of Jordan blocks in each

J

i

is the

dimension

of

W

λ

i

, the eigenspace of

T

corresponding to

λ

i

.

More to come…

References

1

Friedberg, Insell, Spence.

Linear Algebra

. Prentice-Hall Inc., 1997.

Title

generalized eigenspace

Canonical

name

GeneralizedEigenspace

Date of creation

2013-03-22 17:23:36

Last modified on

2013-03-22 17:23:36

Owner

CWoo (3771)

Last modified by

CWoo (3771)

Numerical id

8

Author

CWoo (3771)

Entry type

Definition

Classification

msc 15A18

Related topic

GeneralizedEigenvector

Generated on Fri Feb 9 18:25:46 2018 by

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invariance of eigenvalues

invariance of eigenvalues

Suppose

M

is a

n

×

n

matrix. In the following

proposition

when we say that
“two matrices have the same spectra” we mean that the

eigenvalues

coincide
including

multiplicity

.

1.

If

S

is an

invertible matrix

, then

M

and

S

-

1

⁢

M

⁢

S

have
the same spectra.

Title

invariance of eigenvalues

Canonical

name

InvarianceOfEigenvalues

Date of creation

2013-03-22 15:17:29

Last modified on

2013-03-22 15:17:29

Owner

matte (1858)

Last modified by

matte (1858)

Numerical id

5

Author

matte (1858)

Entry type

Definition

Classification

msc 15A18

Generated on Fri Feb 9 18:25:48 2018 by

LaTeXML
Jordan canonical form theorem

Jordan canonical form theorem

A

Jordan block

or

Jordan matrix

is a matrix of the form

(

λ

1

0

⋯

0

0

λ

1

⋯

0

0

0

λ

⋯

0

⋮

⋮

⋮

⋱

1

0

0

0

⋯

λ

)

with a

constant

value

λ

along the

diagonal

and 1’s on the

superdiagonal

. Some texts the 1’s on the

subdiagonal

instead.

Theorem

.

Let

V

be a

finite-dimensional

vector space

over a field

F

and

t

:

V

→

V

be a

linear transformation

. Then, if the

characteristic polynomial

factors completely over

F

, there will exist a basis of

V

with respect to which the matrix of

t

is of the form

(

J

1

0

⋯

0

0

J

2

⋯

0

⋯

0

0

⋯

J

k

)

where each

J

i

is a Jordan block in which

λ

=

λ

i

.

The matrix in

Theorem

1 is called a

Jordan canonical form

for the

transformation

t

.

Title

Jordan canonical form theorem

Canonical name

JordanCanonicalFormTheorem

Date of creation

2013-03-22 12:59:21

Last modified on

2013-03-22 12:59:21

Owner

Mathprof (13753)

Last modified by

Mathprof (13753)

Numerical id

16

Author

Mathprof (13753)

Entry type

Theorem

Classification

msc 15A18

Synonym

Jordan canonical form

Related topic

PartitionedMatrix

Related topic

SimultaneousUpperTriangularBlockDiagonalizationOfCommutingMatrices

Related topic

Diagonalizable2

Defines

Jordan block

Defines

Jordan matrix

Generated on Fri Feb 9 18:25:50 2018 by

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multiplicity of eigenvalue

multiplicity of eigenvalue

Suppose

V

is a

finite dimensional

vector space

over a field

𝔽

,
and suppose

L

:

V

→

V

is a

linear map

.
Suppose also that

λ

∈

𝔽

is an

eigenvalue

of

L

, that is,

det

⁡

(

L

-

λ

⁢

I

)

=

0

.

The

algebraic multiplicity

,
denoted by

A

λ

⁢

(

L

)

, of

λ

is the

multiplicity

of the root

λ

to the

polynomial

det

⁡

(

L

-

λ

⁢

I

)

=

0

.
The

geometric multiplicity

of

λ

, denoted by

G

λ

⁢

(

L

)

, is the

dimension

of

ker

⁡

(

L

-

λ

⁢

I

)

, the

eigenspace

of

λ

.

Title

multiplicity of eigenvalue

Canonical

name

MultiplicityOfEigenvalue

Date of creation

2013-03-22 15:15:15

Last modified on

2013-03-22 15:15:15

Owner

matte (1858)

Last modified by

matte (1858)

Numerical id

5

Author

matte (1858)

Entry type

Definition

Classification

msc 15A18

Defines

geometric multiplicity

Defines

algebraic multiplicity

Generated on Fri Feb 9 18:26:02 2018 by

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Perron-Frobenius theorem

Perron-Frobenius theorem

Let

A

be a

nonnegative matrix

. Denote its

spectrum

by

σ

⁢

(

A

)

.
Then the

spectral radius

ρ

⁢

(

A

)

is an

eigenvalue

, that is,

ρ

⁢

(

A

)

∈

σ

⁢

(

A

)

, and is associated to a nonnegative

eigenvector

.

If, in addition,

A

is an

irreducible matrix

, then

|

ρ

⁢

(

A

)

|

≥

|

λ

|

, for all

λ

∈

σ

⁢

(

A

)

,

λ

≠

ρ

⁢

(

A

)

, and

ρ

⁢

(

A

)

is a simple eigenvalue associated to a positive eigenvector.

If, in addition,

A

is a

primitive matrix

, then

ρ

⁢

(

A

)

>

|

λ

|

for all

λ

∈

σ

⁢

(

A

)

,

λ

≠

ρ

⁢

(

A

)

.

Title

Perron-Frobenius theorem

Canonical name

PerronFrobeniusTheorem

Date of creation

2013-03-22 13:18:26

Last modified on

2013-03-22 13:18:26

Owner

jarino (552)

Last modified by

jarino (552)

Numerical id

5

Author

jarino (552)

Entry type

Theorem

Classification

msc 15A18

Related topic

FundamentalTheoremOfDemography

Generated on Fri Feb 9 18:26:04 2018 by

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proof of calculus theorem used in the Lagrange method

proof of calculus theorem used in the Lagrange method

Let

f

⁢

(

𝐱

)

and

g

i

⁢

(

𝐱

)

,

i

=

0

,

…

,

m

be

differentiable

scalar functions;

𝐱

∈

R

n

.

We will find local extremes of the

function

f

⁢

(

𝐱

)

where

∇

⁡

f

=

0

. This can be proved by

contradiction

:

∇

⁡

f

≠

0

⇔

∃

ϵ

0

>

0

,

∀

ϵ

;

0

<

ϵ

<

ϵ

0

:

f

(

𝐱

-

ϵ

∇

f

)

<

f

(

𝐱

)

<

f

(

𝐱

+

ϵ

∇

𝐟

)

but then

f

⁢

(

𝐱

)

is not a local extreme.

Now we put up some conditions, such that we should find the

𝐱

∈

S

⊂

R

n

that gives a local extreme of

f

. Let

S

=

⋂

i

=

1

m

S

i

, and
let

S

i

be defined so that

g

i

⁢

(

𝐱

)

=

0

⁢

∀

𝐱

∈

S

i

.

Any vector

𝐱

∈

R

n

can have one

component

perpendicular

to
the subset

S

i

(for visualization, think

n

=

3

and let

S

i

be a flat surface).

∇

⁡

g

i

will be perpendicular to

S

i

, because:

∃

ϵ

0

>

0

,

∀

ϵ

;

0

<

ϵ

<

ϵ

0

:

g

i

⁢

(

𝐱

-

ϵ

⁢

∇

⁡

g

i

)

<

g

i

⁢

(

𝐱

)

<

g

i

⁢

(

𝐱

+

ϵ

⁢

∇

⁡

g

i

)

But

g

i

⁢

(

𝐱

)

=

0

, so any vector

𝐱

+

ϵ

⁢

∇

⁡

g

i

must be outside

S

i

, and also outside

S

.
(todo: I have proved that there might exist a component perpendicular to each subset

S

i

, but not that there exists only one; this should be done)

By the

argument

above,

∇

⁡

f

must be zero - but now we can ignore
all components of

∇

⁡

f

perpendicular to

S

. (todo: this should be expressed more formally and proved)

So we will have a local extreme within

S

i

if there exists a

λ

i

such that

∇

⁡

f

=

λ

i

⁢

∇

⁡

g

i

We will have local extreme(s) within

S

where there exists a set

λ

i

,

i

=

1

,

…

,

m

such that

∇

⁡

f

=

∑

λ

i

⁢

∇

⁡

g

i

Title

proof of

calculus

theorem

used in the Lagrange method

Canonical name

ProofOfCalculusTheoremUsedInTheLagrangeMethod

Date of creation

2013-03-22 13:29:51

Last modified on

2013-03-22 13:29:51

Owner

mathcam (2727)

Last modified by

mathcam (2727)

Numerical id

6

Author

mathcam (2727)

Entry type

Proof

Classification

msc 15A18

Classification

msc 15A42

Generated on Fri Feb 9 18:26:07 2018 by

LaTeXML
proof of Cayley-Hamilton theorem in a commutative ring

proof of Cayley-Hamilton theorem in a commutative ring

Let

R

be a commutative ring with

identity

and let

A

be an order

n

matrix
with elements from

R

⁢

[

x

]

.
For example, if

A

is

(

x

2

+

2

⁢

x

7

⁢

x

2

x

+

1

5

)

then we can also

associate

with

A

the following

polynomial

having matrix coefficents:

A

σ

=

[

0

1

0

5

]

+

[

2

1

0

0

]

⁢

x

+

[

1

0

7

0

]

⁢

x

2

.

In this way we have a mapping

A

⟶

A

σ

which is an

isomorphism

of the rings

M

n

⁢

(

R

⁢

[

x

]

)

and

M

n

⁢

(

R

)

⁢

[

x

]

.

Now let

A

∈

M

n

⁢

(

R

)

and
consider the

characteristic polynomial

of

A

:

p

A

⁢

(

x

)

=

det

⁡

(

x

⁢

I

-

A

)

, which is a

monic
polynomial

of degree

n

with

coefficients

in

R

.
Using a property of the

adjugate matrix

we have

(

x

⁢

I

-

A

)

⁢

adj

⁡

(

x

⁢

I

-

A

)

=

p

A

⁢

(

x

)

⁢

I

.

Now view this as an equation in

M

n

⁢

(

R

)

⁢

[

x

]

. It says that

x

⁢

I

-

A

is a left factor
of

p

A

⁢

(

x

)

. So by the

factor theorem

, the left hand value of

p

A

⁢

(

x

)

at

x

=

A

is 0. The coefficients of

p

A

⁢

(

x

)

have the form

c

⁢

I

, for

c

∈

R

,
so they commute with

A

. Therefore right and left hand values are the same.

References

1

Malcom F. Smiley.

Algebra

of Matrices. Allyn and Bacon, Inc., 1965. Boston, Mass.

Title

proof of

Cayley-Hamilton theorem

in a commutative ring

Canonical name

ProofOfCayleyHamiltonTheoremInACommutativeRing

Date of creation

2013-03-22 16:03:16

Last modified on

2013-03-22 16:03:16

Owner

Mathprof (13753)

Last modified by

Mathprof (13753)

Numerical id

11

Author

Mathprof (13753)

Entry type

Proof

Classification

msc 15A18

Classification

msc 15A15

Generated on Fri Feb 9 18:26:14 2018 by

LaTeXML
proof of Jordan canonical form theorem

proof of Jordan canonical form theorem

This

theorem

can be proved combining the

cyclic decomposition theorem

and the

primary decomposition theorem

.
By

hypothesis

, the

characteristic polynomial

of

T

factorizes completely over

F

, and then so does the

minimal polynomial

of

T

(or its

annihilator polynomial

). This is because the minimal polynomial of

T

has exactly the same factors on

F

⁢

[

X

]

as the characteristic polynomial of

T

. Let’s suppose then that the minimal polynomial of

T

factorizes as

m

T

=

(

X

-

λ

1

)

α

1

⁢

…

⁢

(

X

-

λ

r

)

α

r

.
We know, by the primary decomposition theorem, that

V

=

⊕

i

=

1

r

ker

⁡

(

(

T

-

λ

i

⁢

I

)

α

i

)

.

Let

T

i

be the

restriction

of

T

to

ker

⁡

(

(

T

-

λ

i

⁢

I

)

α

i

)

. We apply now the cyclic decomposition theorem to every

linear operator

(

T

i

-

λ

i

I

)

:

ker

(

T

-

λ

i

I

)

α

i

→

ker

(

T

-

λ

i

I

)

α

i

.

We know then that

ker

(

T

-

λ

i

I

)

α

i

has a basis

B

i

of the form

B

i

=

B

1

,

i

⁢

⋃

B

2

,

i

⁢

⋃

…

⁢

⋃

B

d

i

,

i

such that each

B

s

,

i

is of the form

B

s

,

i

=

{

v

s

,

i

,

(

T

-

λ

i

)

⁢

v

s

,

i

,

(

T

-

λ

i

)

2

⁢

v

s

,

i

,

…

,

(

T

-

λ

i

)

k

s

,

i

⁢

v

s

,

i

}

.

Let’s see that

T

in each of this “cyclic sub-basis”

B

s

,

i

is a

Jordan block

:
Simply notice the following fact about this

polynomials

:

X

⁢

(

X

-

λ

i

)

j

=

(

X

-

λ

i

)

j

+

1

+

X

⁢

(

X

-

λ

i

)

j

-

(

X

-

λ

i

)

j

+

1

=

(

X

-

λ

i

)

j

+

1

+

(

X

-

X

+

λ

i

)

⁢

(

X

-

λ

i

)

j

=

(

X

-

λ

i

)

j

+

1

+

λ

i

⁢

(

X

-

λ

i

)

j

and then

T

⁢

(

T

-

λ

i

⁢

I

)

j

⁢

(

v

s

,

i

)

=

(

T

-

λ

i

)

j

+

1

⁢

(

v

s

,

i

)

+

λ

i

⁢

(

T

-

λ

i

⁢

I

)

j

⁢

(

v

s

,

i

)

.

So, if we also notice that

(

T

-

λ

i

⁢

I

)

k

s

,

i

+

1

⁢

(

v

s

,

i

)

=

0

, we have that

T

in this sub-basis is the Jordan block

(

λ

i

0

0

⋯

0

0

1

λ

i

0

⋯

0

0

0

1

λ

i

⋯

0

0

⋮

⋮

⋮

⋱

⋮

⋮

0

0

0

⋯

λ

i

0

0

0

0

⋯

1

λ

i

)

So, taking the basis

B

=

B

1

⁢

⋃

B

2

⁢

⋃

…

⁢

⋃

B

r

, we have that

T

in this basis has a Jordan form.

This form is unique (except for the order of the blocks) due to the uniqueness of the cyclic

decomposition

.

Title

proof of Jordan canonical form theorem

Canonical

name

ProofOfJordanCanonicalFormTheorem

Date of creation

2013-03-22 14:15:36

Last modified on

2013-03-22 14:15:36

Owner

CWoo (3771)

Last modified by

CWoo (3771)

Numerical id

11

Author

CWoo (3771)

Entry type

Proof

Classification

msc 15A18

Generated on Fri Feb 9 18:26:17 2018 by

LaTeXML
proof of Wielandt-Hoffman theorem

proof of Wielandt-Hoffman theorem

Since both

A

and

B

are normal, they can be diagonalized by

unitary transformations

:

A

=

V

†

⁢

C

⁢

V

and

B

=

W

†

⁢

D

⁢

W

,

where

C

and

D

are

diagonal

,

V

and

W

are unitary, and

(

)

†

denotes the

conjugate transpose

. The

Frobenius matrix norm

is defined by the

quadratic form

∥

A

∥

F

2

=

tr

⁡

[

A

†

⁢

A

]

and is

invariant

under unitary transformations, hence

∥

A

-

B

∥

F

2

=

∥

V

†

⁢

C

⁢

V

-

W

†

⁢

D

⁢

W

∥

F

2

=

∥

C

∥

F

2

+

∥

D

∥

F

2

-

2

⁢

Re

⁡

tr

⁡

[

C

†

⁢

U

†

⁢

D

⁢

U

]

,

where

U

=

W

⁢

V

†

. The matrix

U

is also unitary, let its

matrix elements

be given by

(

U

)

i

⁢

j

=

u

i

⁢

j

. Unitarity implies that the matrix with
elements

|

u

i

⁢

j

|

2

has its row and column sums equal to

1

, in other words,
it is

doubly stochastic

.

The diagonal elements

C

i

⁢

i

=

a

i

are

eigenvalues

of

A

and

D

i

⁢

i

=

b

i

are those of

B

. Writing out the

Frobenius norm

explicitly,
we get

∥

A

-

B

∥

F

2

=

∑

i

(

|

a

i

|

2

+

|

b

i

|

2

)

-

2

⁢

Re

⁢

∑

i

⁢

j

a

¯

i

⁢

|

u

i

⁢

j

|

2

⁢

b

j

≥

∑

i

(

|

a

i

|

2

+

|

b

i

|

2

)

-

2

⁢

min

S

⁡

Re

⁢

∑

i

⁢

j

a

¯

i

⁢

s

i

⁢

j

⁢

b

j

,

where the

minimum

is taken over all

doubly stochastic matrices

S

, whose
elements are

(

S

)

i

⁢

j

=

s

i

⁢

j

. By the Birkoff-von Neumann

theorem

, doubly
stochastic matrices form a closed convex (

http://planetmath.org/ConvexSet

)

polyhedron

with

permutation matrices

at the vertices. The

expression

∑

i

⁢

j

a

¯

i

⁢

s

i

⁢

j

⁢

b

j

is a

linear functional

on this polyhedron,
hence its minimum is achieved at one of the vertices, that is when

S

is a permutation matrix.

If

S

represents the

permutation

σ

, its action can be written as

∑

j

s

i

⁢

j

⁢

b

j

=

b

σ

⁢

(

i

)

. Finally, we can write the last

inequality

as

∥

A

-

B

∥

F

2

≥

∑

i

(

|

a

i

|

2

+

|

b

σ

⁢

(

i

)

|

2

)

-

2

⁢

min

σ

⁡

Re

⁢

∑

i

⁢

j

a

¯

i

⁢

b

σ

⁢

(

i

)

=

min

σ

⁡

|

a

i

-

b

σ

⁢

(

i

)

|

2

,

which is exactly the statement of the

Wielandt-Hoffman theorem

.

Title

proof of Wielandt-Hoffman theorem

Canonical

name

ProofOfWielandtHoffmanTheorem

Date of creation

2013-03-22 14:58:51

Last modified on

2013-03-22 14:58:51

Owner

Andrea Ambrosio (7332)

Last modified by

Andrea Ambrosio (7332)

Numerical id

6

Author

Andrea Ambrosio (7332)

Entry type

Proof

Classification

msc 15A18

Classification

msc 15A42

Generated on Fri Feb 9 18:26:19 2018 by

LaTeXML
spectral values classification

spectral values classification

Spectral points classificationFernando Sanz Gamiz

Definition 1

.

Let

X

a

topological vector space

and

A

:

X

⊃

D

A

⟶

X

a

linear transformation

with
domain

D

A

. Depending on the

properties

of

1

1

the notation

(

λ

-

A

)

is to be
understood as

λ

⁢

I

-

A

with

I

the

identity transformation

and

R

⁢

(

λ

-

A

)

is the range
of

(

λ

-

A

)

(

λ

-

A

)

the following

definitions

apply:

(

λ

-

A

)

-

1

Boundness of

(

λ

-

A

)

-

1

R

⁢

(

λ

-

A

)

Set to which

λ

belongs

exists

bounded

dense in X

resolvent set

ρ

⁢

(

A

)

exists

unbounded

dense in X

continuous spectrum

C

⁢

σ

⁢

(

A

)

exists

bounded or unbounded in X

not dense in X

residual spectrum

R

⁢

σ

⁢

(

A

)

not exists

dense or not dense in X

puntual spectrum

P

⁢

σ

⁢

(

A

)

Remark 1

.

It is

obvious

that, if

F

is the field of possible values for

λ

(usually

F

=

ℂ

or

F

=

ℝ

) then

F

=

ρ

⁢

(

A

)

∪

C

⁢

σ

⁢

(

A

)

∪

R

⁢

σ

⁢

(

A

)

∪

P

⁢

σ

⁢

(

A

)

, that is, these
definitions cover all the possibilities for

λ

. The complement of the resolvent set is called

spectrum

of the operator A, i.e.,

σ

⁢

(

A

)

=

C

⁢

σ

⁢

(

A

)

∪

R

⁢

σ

⁢

(

A

)

∪

P

⁢

σ

⁢

(

A

)

Remark 2

.

In the finite dimensional case if

(

λ

-

A

)

-

1

exists it must be bounded, since all finite
dimensional linear mappings are bounded. This existence also implies that the range of

(

λ

-

A

)

must be the whole X. So, in the finite dimensional case the only

spectral values

we
can encounter are point spectrum values (

eigenvalues

).

Title

spectral values classification

Canonical

name

SpectralValuesClassification

Date of creation

2013-03-22 18:52:01

Last modified on

2013-03-22 18:52:01

Owner

fernsanz (8869)

Last modified by

fernsanz (8869)

Numerical id

5

Author

fernsanz (8869)

Entry type

Definition

Classification

msc 15A18

Synonym

eigenvalues

Synonym

spectrum

Related topic

Eigenvalue

Related topic

SpectrumOfAMuI

Related topic

InvertibleLinearTransformation

Defines

spectrum

Defines

point spectrum

Defines

residual spectrum

Defines

continuous spectrum

Defines

resolvent set

Defines

eigenvalues

Defines

puntual spectrum

Defines

point spectral value

Defines

residual spectral value

Defines

continuous spectral value

Defines

resolvent set value

Generated on Fri Feb 9 18:26:28 2018 by

LaTeXML
spectrum of A-μ⁢I

spectrum of

A

-

μ

⁢

I

Let

A

be an

endomorphism

of the

vector space

V

over a field

k

. Denote by

σ

⁢

(

A

)

the

spectrum

of

A

. Then we have:

Theorem 1

.

σ

⁢

(

A

-

μ

⁢

I

)

=

{

λ

-

μ

:

λ

∈

σ

⁢

(

A

)

}

Theorem

1

is

equivalent

to:

Theorem 2

.

λ

is a

spectral value

of

A

if and only if

λ

-

μ

is a spectral value of

A

-

μ

⁢

I

.

Proof of Theorem

2

.

Note that

A

-

λ

⁢

I

=

(

A

-

μ

⁢

I

)

-

(

λ

⁢

I

-

μ

⁢

I

)

=

(

A

-

μ

⁢

I

)

-

(

λ

-

μ

)

⁢

I

and thus

A

-

λ

⁢

I

is

invertible

if and only if

(

A

-

μ

⁢

I

)

-

(

λ

-

μ

)

⁢

I

is invertible. Equivalently,

λ

is a spectral value of

A

iff

λ

-

μ

is a
spectral value of

(

A

-

μ

⁢

I

)

, as desired.
∎

Title

spectrum of

A

-

μ

⁢

I

Canonical name

SpectrumOfAmuI

Date of creation

2013-03-22 15:32:49

Last modified on

2013-03-22 15:32:49

Owner

PrimeFan (13766)

Last modified by

PrimeFan (13766)

Numerical id

9

Author

PrimeFan (13766)

Entry type

Theorem

Classification

msc 15A18

Related topic

SpectralValuesClassification

Generated on Fri Feb 9 18:26:30 2018 by

LaTeXML
companion matrix

companion matrix

Given a

monic polynomial

p

⁢

(

x

)

=

x

n

+

a

n

-

1

⁢

x

n

-

1

+

…

+

a

1

⁢

x

+

a

0

the

companion matrix

of

p

⁢

(

x

)

, denoted

𝒞

p

⁢

(

x

)

, is the

n

×

n

matrix with

1

’s down the first

subdiagonal

and minus the

coefficients

of

p

⁢

(

x

)

down the last column, or alternatively, as the

transpose

of this matrix. Adopting the first convention this is simply

𝒞

p

⁢

(

x

)

=

(

0

0

…

…

…

-

a

0

1

0

…

…

…

-

a

1

0

1

…

…

…

-

a

2

0

0

⋱

⋮

⋮

⋮

⋱

⋮

0

0

…

…

1

-

a

n

-

1

)

.

Regardless of which convention is used the

minimal polynomial

(

http://planetmath.org/MinimalPolynomialEndomorphism

) of

𝒞

p

⁢

(

x

)

equals

p

⁢

(

x

)

, and the

characteristic polynomial

of

𝒞

p

⁢

(

x

)

is just

(

-

1

)

n

⁢

p

⁢

(

x

)

. The

(

-

1

)

n

is needed because we have defined the characteristic polynomial to be

det

⁡

(

𝒞

p

⁢

(

x

)

-

x

⁢

I

)

. If we had instead defined the characteristic polynomial to be

det

⁡

(

x

⁢

I

-

𝒞

p

⁢

(

x

)

)

then this would not be needed.

Title

companion matrix

Canonical name

CompanionMatrix

Date of creation

2013-03-22 13:17:12

Last modified on

2013-03-22 13:17:12

Owner

aoh45 (5079)

Last modified by

aoh45 (5079)

Numerical id

7

Author

aoh45 (5079)

Entry type

Definition

Classification

msc 15A21

Generated on Fri Feb 9 18:26:32 2018 by

LaTeXML
diagonalizable

diagonalizable

A

square matrix

over any field is

diagonalizable

if it is

similar

to
a

diagonal matrix

. Sometimes such matrices are said to be

diagonable

.

Title

diagonalizable

Canonical name

Diagonalizable

Date of creation

2013-03-22 16:01:05

Last modified on

2013-03-22 16:01:05

Owner

Mathprof (13753)

Last modified by

Mathprof (13753)

Numerical id

8

Author

Mathprof (13753)

Entry type

Definition

Classification

msc 15A21

Synonym

diagonalizable matrix

Synonym

diagonalizable matrices

Related topic

diagonalization

Related topic

JordanCanonicalFormTheorem

Defines

diagonable

Generated on Fri Feb 9 18:26:33 2018 by

LaTeXML
eigenvalues of an involution

eigenvalues of an involution

Proof.

For the first claim suppose

λ

is an

eigenvalue

corresponding to an

eigenvector

x

of

A

. That is,

A

⁢

x

=

λ

⁢

x

.
Then

A

2

⁢

x

=

λ

⁢

A

⁢

x

, so

x

=

λ

2

⁢

x

. As an eigenvector,

x

is non-zero, and

λ

=

±

1

. Now property (1) follows since the

determinant

is
the product of the eigenvalues. For property (2), suppose that

A

-

λ

⁢

I

=

-

λ

⁢

A

⁢

(

A

-

1

/

λ

⁢

I

)

, where

A

and

λ

are as above.
Taking the determinant of both
sides, and using part (1), and the properties of the determinant, yields

det

⁡

(

A

-

λ

⁢

I

)

=

±

λ

n

⁢

det

⁡

(

A

-

1

λ

⁢

I

)

.

Property (2) follows.

□

Title

eigenvalues of an

involution

Canonical name

EigenvaluesOfAnInvolution

Date of creation

2013-03-22 13:38:57

Last modified on

2013-03-22 13:38:57

Owner

Koro (127)

Last modified by

Koro (127)

Numerical id

4

Author

Koro (127)

Entry type

Proof

Classification

msc 15A21

Generated on Fri Feb 9 18:26:35 2018 by

LaTeXML