What are the Properties of (0,1)-Matrices with Constant Row Sum 3?

[eulersolid]

which are the the simpliest properties of a (0,1)-matrix with constant row sum 3

*this matrix is a matrix D which dirives from a linear system of the form

Ci=Xi+Xai+Xbi ai,bi Ε {1,2,..,k} , i=1,2,...,k

or C=DX in the formal language of matrices, thus

D=[dij]=[δ(i,j)+δ(ai,j)+δ(bi,j)] where δ is Kronecker delta

So the Question is detD=0 <=> properties?

 

[Stephen Tashi]

I can't interpret your notation. Perhaps you should give an example.

 

[eulersolid]

Thank you for your answer, my question was very unstable and i was focusing for matrices with aij E{0,1}
My question is a little more general

Those matrices I'm searching for, are in generally, k*k integer matrices A=[aij] (aij integers) with constant row sum 3
(and why not, in the more general case; constant row sum s but this would be in another conversation)

for example the matrix

0 1 2
1 1 1
1 1 1

In the spesial case of 3*3 nteger matrices A=[aij] (aij integers) with constant row sum 3

DetA=0 <=> 2 columns or 2 rows are identical
(thus, the elements aij of this matrix A such that DetA=0 are satisfiing those properties)

My question is; Are there any similar or simple properties for the general case k*k?

I will give you another 2 examples

Thanks a lot

 

[eulersolid]

0 1 2
1 2 0
1 1 1

 

[eulersolid]

1 1 1 0
3 0 0 0
1 2 0 1
0 3 0 0

 

[Stephen Tashi]

I don't know what sort of properties interest you. You are saying that they have the property that their determinant is 0. Since they have determinant 0, the matrices have no inverses. The product of two of them has determinant 0.

They are a scalar multiple of a transition matrix for a Markov process. For example:

$$(1/3) \begin{pmatrix} 0 & 1 & 2 \\1 & 2 & 0 \\1&1&1 \end{pmatrix}$$

can be regarded as a process where the transition probabilities are
0 for state 1 -> state 1
1/3 for state 1 -> state 2
2/3 for state 1 -> state 3
etc.

 

[eulersolid]

let A=[aij],( aij integers) with row sum 3
, i would be interest to know if there are any theorem which says weather a matrix of this form has DetA=0
If you know some familiar theorem please reply
Thanks a lot

 

[Stephen Tashi]

$$\begin{pmatrix} 1 & 1 & 1 \\1 & 2 & 0 \\2 & 1 & 0 \end{pmatrix}$$
doesn't have determinant = 0.

 

[eulersolid]

i know that, those matrices i presented where examples of integer matrices which have row sum 3 *i didnt say they have determinant 0

 

[Stephen Tashi]

let A=[aij],( aij integers) with row sum 3
, i would be interest to know if there are any theorem which says weather a matrix of this form has DetA=0

No, there is no theorem that says such matrices always have determinant = 0.

i know that, those matrices i presented where examples of integer matrices which have row sum 3 *i didnt say they have determinant 0

OK, if you know that they don't always have determinant = 0, then what exactly are you asking? It sounds like you want to know some special condition on these matrices that would imply they were invertible? You don't want the conditions that would work for all matrices such as being of full row rank?

 

[eulersolid]

my question is general I am not searchin for 3*3 matrices, but for k*k matrices

I believe that the following phrase is true, but i can't prove it

Let A be an integer k*k matrix with row sum 3

Then DetA equals to 0

if and only if

A is trivial(two columns or two rows of the matrix are identical)

Is this true ?

 

[eulersolid]

this not an obvious phrase
if DetA equals 0 , that means that A has two rows or two columns identical the same?
Its difficult to tell

 

[Stephen Tashi]

this not an obvious phrase
if DetA equals 0 , that means that A has two rows or two columns identical the same?
Its difficult to tell

Let A be a 3 by 3 matrix whose entries are non-negative integers and such that the sum of the entries in each row is 3. If Det(A) = 0 then must at least two rows or two columns of A be identical?

Answer: No. For example:

$$\begin{pmatrix} 0 & 0 & 3 \\ 0 & 2 & 1 \\ 0 & 1 & 2 \end{pmatrix}$$

 

[eulersolid]

i meant
beside this obvius cases; some column or some row is the zero column or the zero row
s
so i will rephraze

I believe that the following phrase is true, but i can't prove it

Let A be an integer k*k matrix with row sum 3

Then DetA equals to 0

if and only if

A is trivial

or A=BC where B or C is trivial


(trivial: some column is the zero column or some row is the zero row or two columns or two rows of the matrix are identicaly the same)

Is this true ?

 

[Stephen Tashi]

OK, let's try this version of your questions:
----------------
Define a "trivial" matrix to be a matrix A which satisfies at least one of the following conditions:
1. At least two rows of A are equal, as vectors.
2. At least two columns of A are equal as vectors
3. At least one row or one column of A is entirely zeros

Let C be collection of 3 by 3 matrices whose entries are non-negative integers and such that each row sum in the matrix is 3.

Is the following true?
if A is a member of C and Det(A) = 0 then A is trivial?

Would a similar statement be true for larger matrices, such as 4x4, 5x5 etc.?

----------------

Is that an accurate statement of what you are asking? You put "(0,1)" in the title of your post. Are we assuming that 2 and 3 can also be entries in the matrices?


I don't know of any theorem that is relevant. Have you tried investigating numerical examples with a computer program?

With your permission, I'll start a new thread about this question. This one may be too long to attract the attention of people who might know how to contribute.

 

[eulersolid]

integer matrix with constant row sum 3

My friend thank you for your interest
I will give you one phrase for the simplier case of 2 which can easy be proved

G={A=[aij] an integer matrix, aijEZ with constant row sum 2}.
Let AEG Then

DetA=0

iff

There exist B,C EG such that A=B*C
where B is a cyclic matrix

1100...00
0110...00
.... ,with rankB=2s=even (properties)
0000...11
1000...01

 

[eulersolid]

My friend thank you for your interest, i have no problem for that
theremight be more properties for those matrics having det=0 than the obvius ones
I will give you one phrase for the simplier case of 2 which can easy be proved

G={A=[aij] an integer matrix, aijEZ with constant row sum 2}.
Let AEG with 1s or 2s or 3s in the diagonial Then

DetA=0

iff

There exist B,C EG such that A=B*C * means times...
where B is a cyclic matrix

1100...00
0110...00
.... ,with rankB=2s=even (properties)
0000...11
1000...01

the obvius cases (equal columns or equal rows are a spesial case of cuclic matrix of rank 2)

11
11

 

[Stephen Tashi]

I am confused about which set of integer you want to allow as entries. Is the main question about matrices whose integers entries are only: { 0, 1} ? Or, in the case of a kxk matrix, do you want to allow the entries {0,1,2,...k}? Or, in the case of a kxk matrix, whose row sums = r, do you want to allow { 0,1,2,..r} ?

 

[Stephen Tashi]

Did you understand my question?

By the way, have you investigated your conjecture with a computer program?

 

[eulersolid]

Yes i will allow those entries{1,...,r} for the general conjecture constant row sum r
Unfortunatelly i don't have the means for such a great search! If you are able to do it i would be very greatfull!
In my first question i asked you if this is true "A has det=0 iff some columns or rows are identical",
i have found a conterexample a 7*7 matrix with constant row sum 3 and no columns or rows the same but with determinant 0!

that matrix is

1 0 0 0 0 2 0
0 1 0 0 1 1 0
0 0 2 0 0 0 1
0 0 0 2 1 0 0
0 0 1 0 2 0 0
0 0 1 1 0 1 0
1 0 0 0 1 0 1

so the properties for those matrices i search for maybe too complex to investigate them
I believe that maybe we must add more properties for those matrices ie A=B*C
B or C have some identical columns or rows or etc

 

[Stephen Tashi]

Let me ask you about this reasoning:

If Det(A) = 0, we can perform "elementary row operations" on A and transform it to a matrix that is trivial, correct? Gaussian elimination should do it. This can be represented by multiplying A on the left by some matrix G.

So GA = B is trivial. The matrix G is invertible. So A = G^(-1) B.

I don't see how it is possible to distinguish an interesting set of matrices by saying that they have a trivial factor.

------------------

I may eventually be able to investigate some things with a computer program, but first I must revive my programming skills since they have been dormant for 3 years, since I retired.