proportions of invertible matrices

proportions of invertible matrices

Let

G

⁢

L

⁢

(

d

,

R

)

denote the

invertible

d

×

d

-matrices over a ring

R

, and

M

d

⁢

(

R

)

the set of all

d

×

d

-matrices over

R

. When

R

is a finite field of order

q

, commonly denoted

G

⁢

F

⁢

(

q

)

or

𝔽

q

, we prefer to write simply

q

. In particular,

q

is a power of a prime.

Proposition 1

.

|

G

⁢

L

⁢

(

d

,

q

)

|

|

M

d

⁢

(

q

)

|

=

∏

i

=

1

d

(

1

-

1

q

i

)

.

Proof.

The number of

d

×

d

-matrices over a

G

⁢

F

⁢

(

q

)

is

q

d

2

. When a matrix
is invertible, its rows form a basis of the

vector space

G

⁢

F

⁢

(

q

)

d

and this
leads to the following formula

|

G

⁢

L

⁢

(

d

,

q

)

|

=

q

(

d

2

)

⁢

∏

i

=

1

d

(

q

i

-

1

)

.

(Refer to order of the

general linear group

. (

http://planetmath.org/OrdersAndStructureOfClassicalGroups

))

Now we prove the ratio holds:

∏

i

=

1

d

(

1

-

1

q

i

)

=

∏

i

=

1

d

q

i

-

1

q

i

=

1

q

(

d

+

1

2

)

⁢

∏

i

=

1

d

(

q

i

-

1

)

=

1

q

d

2

⁢

q

(

d

2

)

⁢

∏

i

=

1

d

(

q

i

-

1

)

=

|

G

⁢

L

⁢

(

d

,

q

)

|

|

M

d

⁢

(

q

)

|

.

∎

Corollary 2

.

As

q

→

∞

with

d

fixed, the

proportion of invertible matrices

to all matrices

converges

to 1. That is:

lim

q

→

∞

⁡

|

G

⁢

L

⁢

(

d

,

q

)

|

|

M

d

⁢

(

q

)

|

=

1

.

Corollary 3

.

As

d

→

∞

and

q

is fixed, the proportion of invertible matrices decreases

monotonically

and converges towards a positive limit. Furthermore,

1

4

≤

1

-

q

2

-

q

+

1

q

2

⁢

(

q

-

1

)

≤

∏

i

=

1

d

(

1

-

1

q

i

)

≤

1

-

1

q

.

Proof.

By direct expansion we find

∏

i

=

1

∞

(

1

-

a

i

)

=

1

-

∑

1

≤

i

a

i

+

∑

1

≤

i

<

j

a

i

⁢

a

j

-

∑

1

≤

i

<

j

<

k

a

i

⁢

a

j

⁢

a

k

+

⋯

.

So setting

a

0

=

1

and

a

i

+

1

=

a

i

⁢

∑

j

=

i

+

1

∞

1

q

j

=

a

i

⁢

1

q

i

⁢

(

q

-

1

)

for all

i

∈

ℕ

, we have

∏

i

=

1

∞

(

1

-

1

q

i

)

=

∑

i

=

0

∞

(

-

1

)

i

⁢

a

i

.

As

a

i

≥

0

,

a

i

≥

a

i

+

1

for all

i

∈

ℕ

and

a

i

→

0

as

i

→

∞

, we may use Leibniz’s theorem to conclude the

alternating
series

converges. Furthermore, we may estimate the error to the

N

-th term with
error within

±

a

N

+

1

. Using

N

=

2

we have an estimate of

1

-

1

/

q

with error

±

1

q

2

⁢

(

q

-

1

)

. Since

q

≥

2

this gives

1

/

2

with error

±

1

/

4

. Thus
we have at least

1

/

4

chance of choosing an

invertible matrix

at random.
∎

Remark 4

.

q

=

2

is the only field size where the proportion of invertible matrices to all matrices is less than

1

/

2

.

Acknowledgements: due to discussions with Wei Zhou, silverfish and mathcam.

Title

proportions of invertible matrices

Canonical name

ProportionsOfInvertibleMatrices

Date of creation

2013-03-22 15:57:35

Last modified on

2013-03-22 15:57:35

Owner

Algeboy (12884)

Last modified by

Algeboy (12884)

Numerical id

16

Author

Algeboy (12884)

Entry type

Theorem

Classification

msc 15A33

Synonym

proportions invertible linear transformations

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