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ai

5

Large spaes of matries with

bounded rank

D r

Clément de Seguins Pazzis

Lyée Sainte-Geneviève, Versailles, Frane

Université de Versailles-Saint-Quentin-en-Yvelines

dsp.profgmail.om

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

I. Historial results.

II. Flanders's method.

III. Proof of the generalized Atkinson-Lloyd

theorem.

IV. Appliation to an invertibility preserver

problem.

K

n

p

with

n ≥ p

M n,p ( K )

with

n

p

K

M n ( K ) := M n,n ( K )

Two linear subspaes

V

W

M n,p ( K )

equivalent when

∃(P, Q) ∈ GL n ( K ) × GL p ( K )

suh that

W = P V Q

The upper rank of a subset

V

M n,p ( K )

rk(V ) := max

rk M | M ∈ V .

Problem : given an integer

r > 0

r < n

r < p

M n,p ( K )

with upper rank

r

For small values of

r

(Shur:

r = 1

2 ≤ r ≤ 3

values, no fully enompassing answer is known.

with upper rank

r

(ii) Classify those subspaes with a dimension

lose to the maximal one.

A. Maximal dimension

Theorem 1 (Flanders, 1962). Let

r ∈ [[1, p − 1]]

and

V

M n ( K )

rk(V ) = r

(a)

dim V ≤ rn

(b) If

dim V = rn

n > p

V

to

h

C ₁ · · · C _r [0] _n ^× _{( p} ⁻ _{r )}

i | C ₁ , . . . , C _r ∈ M n, 1 ( K )

() If

dim V = rn

n = p

V

V ^T

equivalent to

h

C ₁ · · · C _r [0] _n ^× _(n ⁻ _r)

i | C ₁ , . . . , C _r ∈ M n, 1 ( K )

The ase

# K > r

H. Flanders, On spaes of linear transformations with

bounded rank, J. Lond. Math. So. 37 (1962) 10-16.

The ase

n = p

r = n − 1

eld (extended to ane subspaes):

J. Dieudonné, Sur une généralisation du groupe orthogonal

à quatre variables, Arh. Math. 1 (1949) 282-287.

The more general ase:

R. Meshulam, On the maximal rank in a subspae of

matries, Quart. J. Math. Oxford (2) 36 (1985) 225-229.

The more general ase extended to ane

subspaes:

C. de Seguins Pazzis, The ane preservers of non-singular

matries, Arh. Math. 95 (2010) 333-342.

The naive onjeture: a subspae

V

rank

r

dimension

nr

n > p

matries of

V

(p − r)

K ^p

n = p

this should hold for

V

V ^T

True for

r = 1

Notation 1. Given

(s, t) ∈ [[0, n]] × [[0, p]]

denes

R(s, t)

n × p

the form





[?] s × t [?] _s ^× _{( p} ⁻ _{t )} [?] _(n ⁻ _s) ^× _t [0] _(n ⁻ _s) ^× _(p ⁻ _t)



 .

Note that

rk R(s, t)

≤ s + t

whenever

s + t ≤ p

In partiular, if

r ≥ 1

R(1, r − 1)

ounter-example to the above onjeture, with

dimension

n(r − 1) + (p − r + 1) = nr − (n − p + r − 1)

partiular:

nr − (r − 1)

n = p

Atkinson and LLoyd ask: an we lassify spaes

V

r

nr − (n − p + r − 1) ≤ dim V ≤ nr

Theorem 2 (Atkinson and Lloyd, 1980).

Let

r ∈ [[1, n − 1]]

V

M n ( K )

rk(V ) = r

Assume that

# K > r

dim V ≥ nr − r + 1

Then:

(a) Either

V

W

M n ( K )

rk(W ) = r

dim W = nr

(b) Or

V

R(1, r − 1)

R(r − 1, 1)

bounded rank, Quart. J. Math. Oxford (2), 31 (1980)

253-262.

The ase

n > p

L.B. Beasley, Null spaes of spaes of matries of bounded

rank, in Current Trends in Matrix Theory 45-50, Elsevier,

1987.

What about small nite elds?

An extension of Atkinson-Lloyd to an arbitrary

eld was deemed problemati: ounter-example

for

F 2

(









a c d

0 b e

0 0 a + b









| a, b, c, d, e ∈ F 2

)

ts into neither of the above ategories.

Theorem 3 (de Seguins Pazzis, 2010). Atkinson

and Lloyd's theorem holds for an arbitrary eld,

exept when

n = 3

r = 2

# K = 2

dim V = 5

in whih ase the above ounter-example is the

only exeption up to equivalene.

C. de Seguins Pazzis, The lassiation of large spaes of

matries with bounded rank, arXiv:

http://arxiv.org/abs/1004.0298

Beasley's results are also suessfully extended to

an arbitrary eld (no exeption!).

To simplify things, we take

n = p

A. The original method.

Let

V

M n ( K )

rk(V ) = r ∈ [[1, n − 1]]

V

ontains

J _r :=





I _r [0] _r ^× _(n ⁻ _r) [0] _{( n} ⁻ _{r )} ^× _r [0] _{( n} ⁻ _{r )} ^× _{( n} ⁻ _{r )}



 .

In that ase, we split every matrix of

V

same blok sizes:

M =





A(M ) C (M ) B (M ) D(M )



 .

Assume that

# K > r

M ∈ V

an indeterminate

t

(n − r) × (n − r)

matrix

P (t)

(r + 1) × (r + 1)

J _r + tM

r

olumns:

→ P (t)

t

M n − r ( K )

≤ r

→ P (t)

K

P (t) = 0

→

t ^r

D(M )

D(M ) = 0 ;

→

t ^{r − 1}

−B (M )C (M )

B (M )C (M ) = 0.

From

∀M ∈ V, B(M )C (M ) = 0

that

dim B (V ) + dim C (V ) ≤ r(n − r)

B (V ) × C (V )

non-degenerate quadrati form

(B, C ) 7→ tr(BC )

Therefore:

dim V ≤ dim A(V ) + dim B (V ) + dim C (V ) ≤ nr.

Assume now

dim V = nr

V

ontains, for every invertible

P ∈ M n ( K )

M _P :=





P [0] _r ^× _{( n} ⁻ _{r )} [0] _{( n} ⁻ _{r )} ^× _r [0] _{( n} ⁻ _{r )} ^× _{( n} ⁻ _{r )}



 .

With the above method, one replaes

J _r

M _P

and nds:

∀M ∈ V, B (M )P ^{− 1} C (M ) = 0.

Then varying

P

∀M ∈ V, B(M ) = 0

C (M ) = 0.

One easily onludes: either

B (V ) = {0}

C (V ) = {0}

Same starting point, assumptions (exept on the

ardinality of

K

assuming that

V

J _r

that it ontains





Q [0] _r ^× _{( n} ⁻ _{r )} [0] _{( n} ⁻ _{r )} ^× _r [0] _{( n} ⁻ _{r )} ^× _{( n} ⁻ _{r )}





for some invertible

Q ∈ M r ( K )

We take an arbitrary matrix of

M n ( K )

M ₀ =





P C (M ₀ ) B (M ₀ ) D(M ₀ )



 .

We atually do not ompute the matrix of minors

of a general sum

M ₀ + M

M ∈ V

ompliated!). Rather, we onsider a tower of

linear subspaes of

V

V ₃ ⊂ V ₂ ⊂ V ₁ ⊂ V.



 



 



V ₁ := Ker A

V ₂ := Ker A ∩ Ker B

V ₃ = Ker A ∩ Ker B ∩ Ker C.

The rank theorem shows:

dim V = dim A(V )+dim B (V ₁ )+dim C (V ₂ )+dim D(V ₃ ).

One then omputes the matrix of all the

(r + 1) × (r + 1)

M ₀ + M

r

M ∈ V ₁

det(P )D(M ₀ + M )

− B (M ₀ + M ) com(P ) ^T C (M ₀ + M ).

This matrix is

0

M = 0

Subtrating both equalities yields the

fundamental formula:

B (M ) com(P ) ^T c(M ) = det(P )D(M )

− B (M ) com(P ) ^T c(M ₀ )

− B (M ₀ ) com(P ) ^T c(M )

As

P

D(V ₃ ) = 0.

Taking the polar form of the left-hand side yields,

for every

(M, N ) ∈ V ₁ ²

B (M ) com(P ) ^T c(N ) + B (N ) com(P ) ^T c(M ) = 0.

In partiular, for every

(M, N ) ∈ V ₁ × V ₂

B (M ) com(P ) ^T c(N ) = 0.

In partiular

∀(M, N ) ∈ V ₁ × V ₂ , B(M )Q ^{− 1} C (N ) = 0

therefore

dim B (V ₁ ) + dim C (V ₂ ) ≤ r(n − r).

This yields

dim V ≤ nr

For the ase

dim V = nr

the above formulas ...

generalized Atkinson-Lloyd

theorem.

We ontinue with the assumptions of the latest

setion. We now assume

dim V > nr − r + 1

A. Step 1: showing that

B ( V ₁ ) = 0

C ( V ₂ ) = {0}

We show that either there is no non-zero matrix

in

V





[0] r × n

[?] _(n ⁻ _r) ^× _n





or there is no non-zero matrix in

V

h

[0] n × r [?] _n ^× _{( n} ⁻ _{r )} i

.

Using the above inequalies, we nd

dim A(V ) > r ² − (r − 1).

∀P ∈ A(V ), ∀(M, N ) ∈ V ₁ × V ₂ ,

B (M ) com(P ) ^T C (N ) = 0.

Lemma 4. One has

B (V ₁ ) = {0}

C (V ₂ ) = {0}

Proof. Assume the ontrary holds. Then

span

{com(P ) ^T | P ∈ A(V )}

transitively on

K ^r

may then assume that

∀P ∈ A(V ), com(P ) r,r = 0.

This ontradits the Flanders theorem!!

If

B (V ₁ ) = {0}

W := V ^T

C (W ₂ ) ⊂ B (V ₁ ) ^T = {0}

lost by assuming that

C (V ₂ ) = {0}

V ₂ = V ₃ = {0}

We now assume

V ₂ = {0}

linear subspae

W ⊂ M n,r ( K )

ϕ : W → M _n,n ⁻ _r ( K )

V = h

M ϕ(M )

i | M ∈ W

.

Note that

codim W < r − 1

From there, we prove two lemmas.

Lemma 5. One has

∀N ∈ W, Im ϕ(N ) ⊂ Im(N ).

Lemma 6. There exists

C ∈ M r,n − r ( K )

∀N ∈ W, ϕ(N ) = N C

With the last one, take

P :=





I _r −C [0] _{( n} ⁻ _{r )} ^× _r I _n ⁻ _r





and hek that any matrix of

V P

n − r

Proof of Lemma 5: Denote by

U

matries of

W

0

M ∈ U

write

h

M ϕ(M ) i =





K (M ) [?] _(n ⁻ ₁₎ ^× _(n ⁻ _r) [0] ₁ ^× r ϕ _n (K (M )).





Note that:

→ codim K (U ) < r − 1 < n − 2

→

rk K (M ) = r

ϕ n (K (M )) = 0

→ ϕ _n : K (U ) → M _1,n ⁻ r ( K )

We dedue that

ϕ _n = 0

orollary of our extension of Flander's theorem to

ane subspaes:

Corollary 7. Let

V ^′

M n,p ( K )

n ≥ p ≥ r

dim V ^′ > rn

V ^′

of rang

≥ r

n = p = 2

r = 1

# K = 2

a linear hyperplane

H

V ^′

rank

≥ r

V ^′

disjoint ane hyperplane

H ^′

dimension

≥ rn

< r

linear subspae of

M n,r ( K )

Bak to the proof of Lemma 5: we now have

ϕ _n = 0

H = K ^{n − 1} × {0}

∀M ∈ W, Im M ⊂ H ⇒ Im ϕ(M ) ⊂ H

However,

H

hyperplane of

K ⁿ

Lemma 6 may be extended as some kind of linear

preserver theorem:

Theorem 8 (Representation lemma). Let

m, n, p

be positive integers,

W

M m,n ( K )

codim W ≤ m − 2

ϕ : W → M m,p ( K )

∀M ∈ W, Im ϕ(M ) ⊂ Im M.

Then there exists

C ∈ M n,p ( K )

ϕ : M 7→ M C.

Remarks 1.

•

W = M _m,p ( K )

•

p = 1

•

codim W ≤ m − 2

optimal. When

# K > 2

n ≥ 2

upper bound is

2m − 3

The proof is a bit too tehnial for a seminar talk

...

invertibility preserver problem.

The following theorem is the rst appliation of

the Atkinson-Lloyd theorem.

Theorem 9 (de Seguins Pazzis, 2012). Let

V

a linear subspae of

M n ( K )

≤ n − 2

f : V ֒ → M n ( K )

embedding whih is a strong invertibility

preserver. Then, unless

n = 3

# K = 2

codim V = 1

(P, Q) ∈ GL n ( K ) ²

∀M, f (M ) = P M Q

∀M, f (M ) = P M ^T Q

If

K

f (V ) = V

assume that

f

This is a grand generalization of the well-known

Dieudonné theorem (ibid). The method is similar,

with the Atkinson-Lloyd theorem as a key

starting point. The above representation theorem

is also a ruial point.

non-singularity in a large spae of matries, Linear

Algebra Appl. 436-9 (2012), 3507-3530.

Remark 2. In the ase

n > p

rk M = p

M

hold (urrently being written).