# Rank of Matrix A with Kronecker Symbol and Sum Condition

• zed123
In summary, the conversation revolves around a result found while working on a combinatorics problem. The result states that for a real matrix A with certain properties, the rank of A is equal to 2n, where n is a positive integer. The conversation also includes questions and discussions on the properties and examples of such a matrix, as well as a conjecture for even dimensions. There is also mention of using induction and the possibility of solving the original combinatorics problem instead.
zed123
helloo
while working on a combinatorics problem I have found the following result:

let $A=(a_{ij})_{1\leq i,j\leq2n+1}$ where n is a positive integer , be a real Matrix such that :
i) $a_{ij}^2=1-\delta_{ij}$ where $\delta$ is the kronecker symbol
ii) $\forall i \displaystyle{ \sum_{j=1}^{2n+1}a_{ij}=0}$
then $rankA=2n$
any idea ?

Last edited:
Er, what are you asking? Did you mean that you have observed it in some cases, and are wondering if it's true in general?

Can you describe qualitatively what such a matrix looks like?

I feel like induction is the most likely way to go about it, if it is true. How many particular examples have you tested, and of what sizes? Do you have a conjecture for how things behave if the dimension is even instead of odd?

(Or, maybe you could explain the combinatorics problem you were solving; maybe it's easier to do that problem than it is to work with this matrix)

For n= 1, that is saying that
$$A= \begin{bmatrix}0 & 1 & 1 \\ 1 & 0 & 1\\ 1 & 1 & 0\end{bmatrix}$$
What is the rank of that matrix?

HallsofIvy said:
For n= 1, that is saying that
$$A= \begin{bmatrix}0 & 1 & 1 \\ 1 & 0 & 1\\ 1 & 1 & 0\end{bmatrix}$$
What is the rank of that matrix?

Not exactly For n=1, it's a matrix that looks like this

$$A= \begin{bmatrix}0 & 1 & -1 \\ 1 & 0 & -1\\ -1 & 1 & 0\end{bmatrix}$$

So the entries on the diagonal must be 0, and all the other entries are 1 and -1. But the sum of every row must be 0.

It is very easy to see that such a matrix cannot have full rank (the sum of all the columns is 0, so the columns cannot be linear independent). So the rank is at most 2n. That it's exactly 2n is a bit harder...

Based on the information provided, it seems like the matrix A has some special properties. The first condition given, a_{ij}^2=1-\delta_{ij}, indicates that the matrix is likely symmetric and has a diagonal of all 1's except for the main diagonal, which is filled with -1's. This can also be seen as a Hadamard matrix, which has a well-known property of having a rank equal to its order. In this case, the order of A is 2n+1, so the rank would be 2n+1.

The second condition states that the sum of each row of A is equal to 0. This means that the rows of A are linearly dependent, which would decrease the rank. However, since the matrix is symmetric and has a special structure, it is possible to show that the rank is still equal to 2n.

One approach to proving this would be to use the properties of the Kronecker symbol and the sum condition to construct a basis for the null space of A. This basis would consist of vectors that satisfy both conditions and can be used to show that the rank of A is 2n.

In conclusion, the rank of matrix A with the given conditions is 2n, which is equal to its order. This is a interesting result and could potentially have applications in combinatorics and linear algebra. Further investigation and analysis of this matrix could lead to a better understanding of its properties and potential uses.

## 1. What is the rank of a matrix with Kronecker symbol and sum condition?

The rank of a matrix with Kronecker symbol and sum condition can vary depending on the specific values and conditions of the matrix. In general, the rank will be equal to the number of linearly independent rows or columns in the matrix.

## 2. How is the rank of a matrix with Kronecker symbol and sum condition calculated?

The rank of a matrix with Kronecker symbol and sum condition can be calculated using various methods such as Gaussian elimination or singular value decomposition. These methods involve manipulating the matrix to its reduced row echelon form and counting the number of non-zero rows or columns.

## 3. Can the rank of a matrix with Kronecker symbol and sum condition be less than the number of rows or columns?

Yes, the rank of a matrix with Kronecker symbol and sum condition can be less than the number of rows or columns. This means that there are linear dependencies among the rows or columns of the matrix, resulting in a lower rank than the matrix's dimension.

## 4. What is the relationship between the rank of a matrix with Kronecker symbol and sum condition and its determinant?

The rank of a matrix with Kronecker symbol and sum condition is related to its determinant. Specifically, if the determinant of the matrix is non-zero, then the rank will be equal to the matrix's dimension. On the other hand, if the determinant is zero, then the rank will be less than the dimension, indicating linear dependencies among the rows or columns.

## 5. Can the rank of a matrix with Kronecker symbol and sum condition be greater than the number of rows or columns?

No, the rank of a matrix with Kronecker symbol and sum condition cannot be greater than the number of rows or columns. This is because the rank is limited by the number of linearly independent rows or columns in the matrix, which cannot exceed the matrix's dimension.

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