d.fg
ai

April

4

Beyond Gerstenhaber: spaes

of matries with spetral

onditions

Clément de Seguins Pazzis

Lyée Sainte-Geneviève, Versailles

Université de Versailles-Saint-Quentin-en-Yvelines

dsp.profgmail.om

http://dsp.prod.free.fr/reherhe.html

I. The adapted vetor method

II. The diagonal-ompatibility method

III. Beyond nilpotent subspaes

K

(possibly nite),

M n ( K )

square matries with

n

NT n ( K )

stritly upper-triangular matries.

E _i,j

elementary matrix with all entries zero, exept an

entry

1

(i, j )

Two linear subspaes

V

W

M _n ( K )

similar, and one writes

V ≃ W

∃P ∈ GL n ( K )

W = P V P ^{− 1}

subspae of

M _n ( K )

elements are. Example: linear subspaes of

NT _n ( K )

Problem : an one lassify the nilpotent

subspaes of

M n ( K )

Bakground: Engel's theorem: every Lie

subalgebra of nilpotent matries of

M n ( K )

Is every nilpotent subspae of

M _n ( K )

triangularizable? YES for

n = 2

n > 2

lassial ounterexample: the matries of the form









0 0 a

0 0 b

−b a 0









are all nilpotent; no ommon eigenvetor!

Gerstenhaber's theorem (1958)

If

V

M _n ( K )

dim V ≤ n(n − 1)

2

and equality holds i

V ≃ NT n ( K )

M. Gerstenhaber, On nilalgebras and linear varieties of

nilpotent matries (I), Amer. J. Math. 80 (1958)

614-622.

Gerstenhaber requires

# K ≥ n

eld, proof ompleted by V.N. Serezhkin:

V.N. Serezhkin, Linear transformations preserving

nilpoteny (in Russian), Izv. Akad. Nauk BSSR, Ser.

Fiz.-Mat. Nauk 125 (1985) 46-50.

Simplied proof for:

•

•

# K > 2

in

B. Mathes, M. Omladi£, H. Radjavi, Linear spaes of

nilpotent matries, Linear Algebra Appl. 149 (1991)

215-225.

An appliation of Gerstenhaber's theorem:

Nilpoteny preservers.

E.P. Botta, S. Piere, W. Watkins, Linear transformations

that preserve nilpotent matries, Pa. J. Math. 104

(1983) 39-46.

Motivation: solve the ase of equality. Let

V

nilpotent subspae of

M n ( K )

V

K ⁿ

One needs a basis

(f ₁ , . . . , f _n )

K ⁿ

every element of

V

Basi idea: rst, nd an appropriate

f _n

should be no rank

1

V

spae

K f _n

Denition 1. A (non-zero) vetor

x ∈ K ⁿ

V

1

V

olumn spae

K x

A. Existene of adapted vetors

Lemma 1. Every nilpotent subspae has an

adapted vetor.

Even better:

Lemma 2. Let

V

M _n ( K )

Then, one of the vetors of the standard basis

(e ₁ , . . . , e _n )

V

Proof by indution on

n

n = 1

Assume

n ≥ 2

is

V

U

V

onsisting of matries with last row zero, and

write every

M ∈ U

M =





K (M ) C (M ) [0] ₁ ^× _(n ⁻ ₁₎ 0



 .

Thus,

K (U )

M n − 1 ( K )

By indution, some

e _i

1 ≤ i ≤ n − 1

K (U )

e ₁

K (U )

e ₁

is not

V

R ₀ ∈ M _1,n ( K ) \ {0}

s.t.





R ₀ [0] _(n ⁻ ₁₎ ^× _n



 ∈ V

R ₀ = h

0 · · · 0 ? i

sine

e ₁

K (U )

Then,

E _1,n ∈ V

More generally, we have some

i 6= n

E _i,n ∈ V

More generally, for every

k ∈ [[1, n]]

f (k) ∈ [[1, n]] \ {k}

E _{f (k),k} ∈ V

One hooses an

f

(i ₁ , . . . , i _p )

i ₁ , . . . , i _p

distint in

[[1, n]]

f (i ₁ ) = i ₂ , . . . , f (i _p ⁻ ₁ ) = i _p

and

f (i _p ) = i ₁

Then,

p

P

k=1

E _{f (i k ),i k}

V

B. Using adapted vetors to prove the

inequality statement

Without loss of generality,

e _n

V

M ∈ V

M =





K (M ) C (M ) L(M ) a(M )





with

K (M )

(n − 1) × (n − 1)

Dene

W

M ∈ V

M =





K (M ) [0] _(n ⁻ ₁₎ ^× ₁ L(M ) a(M )



 .

Then,

K (W )

M n − 1 ( K )

As

e _n

V

dim K (W ) = dim W .

Thus,

dim V = dim C (V ) + dim K (W ).

By indution

dim V ≤ (n − 1) + (n − 1)(n − 2)

2 = n(n − 1)

2 ·

Remark 1. By indution, one an nd a

permutation matrix

P

P V P ^{− 1}

non-zero lower-triangular matrix!

Proposition 3. Let

V

M n ( K )

eigenvalue. Then

V

Corollary 4. Let

V

M _n ( K )

eigenvalue. Then

dim V ≤ ⁿ⁽ⁿ ₂ ^{− 1)} ·

Similar proofs as for nilpotent subspaes.

Corollary 5. Let

V

invertible matries of

M _n ( K )

dim V ≤ ⁿ⁽ⁿ ₂ ^{− 1)} ·

Proof. Denote by

V

V

I _n ∈ V

αI _n − M = α(I _n − α ^{− 1} M )

M ∈ V

α ∈ K \ {0}

No matrix of

V

C. de Seguins Pazzis, On the matries of given rank in a

large subspae, Linear Algebra Appl. 435-1 (2011)

Proposition 6. Let

V

M _n ( K )

≤ 1

in

K

Then

V

n = 2

char( K ) = 2

Theorem 7. With the same assumptions,

dim V ≤ 1 + ⁿ⁽ⁿ ₂ ^{− 1)} ·

Counter-example:

sl ₂ ( K )

char( K ) = 2

C. de Seguins Pazzis, Spaes of matries with a sole

eigenvalue, Lin. Multilin. Alg. 60 (2012) 1165-1190.

onsiders matries with rank

1

Proposition 8. Let

V

M n ( K )

≤ 2

in

K

If

n > 2

char( K ) 6= 2

V

vetor.

Theorem 9. With the same assumptions,

dim V ≤ 2 + ⁿ⁽ⁿ ₂ ^{− 1)} ·

C. de Seguins Pazzis, Spaes of matries with few

eigenvalues, arXiv preprint.

Remark 2. For the at most

3

may fail (as for at most

2

char( K ) = 2

method

A. Setup

Aim: now that we have found an adapted vetor,

ontinue the analysis of the ase of equality.

Hypotheses:

V

n(n − 1)

2

e _n

as in the proof of inequality. Then,

dim C (V ) = n−1

dim K (W ) = (n − 1)(n − 2)

2 ·

By indution

K (W ) ≃ NT n − 1 ( K )

rst

n − 1

K (W ) = NT n − 1 ( K ).

Then, one proves:

Lemma 10.

e ₁

V ^T

Proof. Let

C ∈ M _n,1 ( K )

h

C [0] _n ^× _(n ⁻ ₁₎

i ∈ V

K (W ) = NT _n ⁻ ₁ ( K )

C =





[0] _(n ⁻ ₁₎ ^× ₁ α



 .

As

e _n

V

α = 0

Then, one use a new splitting for

V

M =





a ^′ (M ) L ^′ (M ) C ^′ (M ) K ^′ (M )





with

K ^′ (M )

(n − 1) × (n − 1)

Dene

W ^′

M ∈ V

L ^′ (M ) = 0

• K ^′ (W ^′ )

M n − 1 ( K )

•

e ₁

V ^T

dim K ^′ (W ) = ^{(n − 1)(n} ₂ ^{− 2)} ·

By indution,

K ^′ (W ^′ ) ≃ NT n − 1 ( K ).

Lemma 11. The last vetor of the standard basis

is

K ^′ (W ^′ )

Proof. Let

L ∈ M _1,n ⁻ ₁ ( K )

K ^′ (W ^′ )





[0] _(n ⁻ ₂₎ ^× _(n ⁻ ₁₎ L





C ∈ M _n,1 ( K )





C [0] _(n ⁻ ₁₎ ^× _(n ⁻ ₁₎

? L



 ∈ V

Then

C = 0

K (W ) = NT _n ⁻ ₁ ( K )

L = 0

e _n

V

Lemma 12 (Diagonal-ompatibility). There is a

matrix

Q =





I _n ⁻ ₂ [0] _(n ⁻ ₂₎ ^× ₁ [?] ₁ ^× _(n ⁻ ₂₎ 1





K ^′ (W ^′ ) = Q NT _n ⁻ ₁ ( K )Q ^{− 1}

Main ideas:

Point 1:

No generality is lost in assuming that

Qe _n ⁻ ₁ = e _n ⁻ ₁

e _n ⁻ ₁

is

K ^′ (W ^′ )

Point 2:

For every

U ∈ NT n − 2 ( K )





U [0] _(n ⁻ ₂₎ ^× ₁ [?] ₁ ^× _(n ⁻ ₂₎ 0



 ∈ K ^′ (W ^′ ).

Follows from

K (W ) = NT _n ⁻ ₁ ( K )

For every

L ∈ M _1,n ⁻ ₂ ( K )

V

A _L =









0 L 0

[0] _(n ⁻ ₂₎ ^× ₁ [0] _(n ⁻ ₂₎ ^× _(n ⁻ ₂₎ [0] _(n ⁻ ₂₎ ^× ₁

f (L) ϕ(L) 0









(this uses

K (W ) = NT _n ⁻ ₁ ( K )

For every

C ∈ M _n ⁻ _2,1 ( K )

V

B _C =









0 [0] ₁ ^× _(n ⁻ ₂₎ 0 ψ (C ) [0] _(n ⁻ ₂₎ ^× _(n ⁻ ₂₎ C

g(C ) [0] ₁ ^× _(n ⁻ ₂₎ 0









(this uses

K ^′ (W ^′ ) = NT _n ⁻ ₁ ( K )

For every

U ∈ NT n − 2 ( K )

V

E _U =









0 [0] ₁ ^× _(n ⁻ ₂₎ 0

[0] _(n ⁻ ₂₎ ^× ₁ U [0] _(n ⁻ ₂₎ ^× ₁ h(U ) [0] ₁ ^× _(n ⁻ ₂₎ 0









.

Finally,

V

J =









? [0] ₁ ^× _(n ⁻ ₂₎ 1

[?] _(n ⁻ ₂₎ ^× ₁ [?] _(n ⁻ ₂₎ ^× _(n ⁻ ₂₎ [0] _(n ⁻ ₂₎ ^× ₁

? [?] ₁ ^× _(n ⁻ ₂₎ ?







 .

Then, one suessively proves:

•

λ

ϕ(L) = λL

ψ(C ) = −λC

L

C

•

V

P ^{− 1} V P

P =









1 [0] ₁ ^× _(n ⁻ ₂₎ 0

[0] _(n ⁻ ₂₎ ^× ₁ I _n ⁻ ₂ [0] _(n ⁻ ₂₎ ^× ₁ λ [0] ₁ ^× _(n ⁻ ₂₎ 1









,

and thus one an assume

λ = 0

•

f = 0

g = 0

h = 0

•

V

E _1,n

J

One onludes that

V

NT _n ( K )

ompletes the proof.

C. de Seguins Pazzis, On Gerstenhaber's theorem for

spaes of nilpotent matries over a skew eld, Linear

Algebra Appl. 438-11 (2013) 4426-4438.

III. Beyond nilpotent subspaes

The adapted vetor method and the

diagonal-ompatibility methods yield theorems of

the same avor as Gerstenhaber's.

A. Large spaes of matries with no

non-zero eigenvalue

A matrix

P ∈ GL n ( K )

X ^T P X 6= 0

X ∈ K ⁿ

matries

P

Q

there is a non-singular

R

λ

s.t.

P = λRQR ^T

ϕ

ψ

are similar when there is a non-zero salar

λ

suh that

ψ

λ ϕ

A _n ( K )

n × n

matries.

Theorem 13 (de Seguins Pazzis (2010)). Let

V

be a linear subspae of

M _n ( K )

n(n − 1)

2

eigenvalue. Assume that

# K > 2

non-isotropi matries

P ₁ , . . . , P _p

GL n ₁ ( K ), . . . , GL n _p ( K )

V

spae of all matries of the form











P ₁ A ₁ [?]

.

.

.

[0] P _p A _p











with

A ₁ ∈ A n ₁ ( K ), . . . , A p ∈ A n _p ( K )

P ₁ , . . . , P _p

V

sim-ongruene.

Thus, one is redued to the lassiation of

non-isotropi bilinear forms up to similarity.

C. de Seguins Pazzis, Large ane spaes of non-singular

matries, Trans. Amer. Math. So. 365 (2013)

Theorem 14 (de Seguins Pazzis (2010)). Let

V

be an ane subspae of

M n ( K )

n(n − 1)

2

non-singular. Assume

# K > 2

Then, there are non-isotropi matries

P ₁ , . . . , P _p

respetively in

GL _{n 1} ( K ), . . . , GL _{n p} ( K )

V

form

I _n +











P ₁ A ₁ [?]

.

.

.

[0] P _p A _p











with

A ₁ ∈ A _{n 1} ( K ), . . . , A _p ∈ A _{n p} ( K )

The quadrati forms

X 7→ X ^T P ₁ X

X 7→ X ^T P _p X

V

similarity.

For a generalization to ane spaes of matries

with a lower bound on the rank:

C. de Seguins Pazzis, Large ane spaes of matries with

rank bounded below, Linear Algebra Appl. 437-2 (2012)

most one eigenvalue

Theorem 15. Let

V

M n ( K )

1

K

dim V = 1 + ⁿ⁽ⁿ ₂ ^{− 1)} ·

V ≃ K I _n ⊕ NT n ( K )

situations:

• char( K ) = 2

n ∈ {2, 4}

• char( K ) = 3

n = 3

In the exeptional ases stated above, all the

solutions are known.

C. de Seguins Pazzis, Spaes of matries with a sole

eigenvalue, op.it.

Remark 3. If

char( K ) 6 | n

onsequene of Gerstenhaber's theorem.

Conjeture 1. Let

V

M _n ( K )

≤ 1

in

K

dim V = 1 + ⁿ⁽ⁿ ₂ ^{− 1)} ·

n ≥ 5

and

# K > 2

V = K I _n ⊕ H

matrix of

H

K

C. Large spaes of matries with at

most two eigenvalues

For

char( K ) 6= 2

n ≥ 3

of spaes of matries of

M _n ( K )

≤ 2

eigenvalues in

K

2 + ⁿ⁽ⁿ ₂ ^{− 1)}

C. de Seguins Pazzis, Spaes of matries with few

eigenvalues, op.it.

→

→

properties on matrix spaes with spetral

onditions.

→

proofs!