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I am preparing to publish an academic article on computational efficiency and image processing. In my work, I have come across what I can best describe as a non-square skew (symmetric or repeating) matrix (I know it can't be symmetric since it's non-square).

Here are some examples of what it may look like:

(9 x 2)

\begin{array}{cc} 0 & -6 \\ 0 & -6 \\ 0 & -6 \\ 3 & -3 \\ 3 & -3 \\ 3 & -3 \\ 6 & 0 \\ 6 & 0 \\ 6 & 0 \end{array}

(16 x 3)

\begin{array}{ccc} 0 & -12 & -24 \\ 0 & -12 & -24 \\ 0 & -12 & -24 \\ 0 & -12 & -24 \\ 8 & -4 & -16 \\ 8 & -4 & -16 \\ 8 & -4 & -16 \\ 8 & -4 & -16 \\ 16 & 4 & -8 \\ 16 & 4 & -8 \\ 16 & 4 & -8 \\ 16 & 4 & -8 \\ 24 & 12 & 0 \\ 24 & 12 & 0 \\ 24 & 12 & 0 \\ 24 & 12 & 0 \end{array}

(4 x 6)

\begin{array}{cccccc} 0 & -2 & -4 & -6 & -8 & -10 \\ 0 & -2 & -4 & -6 & -8 & -10 \\ 10 & 8 & 6 & 4 & 2 & 0 \\ 10 & 8 & 6 & 4 & 2 & 0 \end{array}

Is there a specific name for this type of matrix? If so, I could not find one.

Also, what are some properties of this matrix that I may be overlooking?

1) It seems that the rank will always be 2.

2) (If the matrix is A): AA' and A'A is always symmetric.

Thank you for your time.

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3  
Just a note: 2) is true for all matrices. –  EuYu Jan 15 '13 at 15:21
1  
It seems like you could describe your matrices using block matrix notation. That could help you ... –  kjetil b halvorsen Jan 15 '13 at 16:06
1  
What you could do is write it as a Kronecker product: $$ \begin{pmatrix} 0 & -6 \\ 0 & -6 \\ 0 & -6 \\ 3 & -3 \\ 3 & -3 \\ 3 & -3 \\ 6 & 0 \\ 6 & 0 \\ 6 & 0 \end{pmatrix}= 3\begin{pmatrix} 0 & -2 \\ 1 & -1 \\ 2 & -0 \end{pmatrix}\otimes \begin{pmatrix} 1 \\ 1 \\ 1 \end{pmatrix} $$ and maybe use something like Toeplitz matrices... –  draks ... Jan 15 '13 at 16:29
    
It looks like the rows and columns are all arithmetic progressions. Using the representation draks has mentioned above, I think your matrices might be of the form $r A \otimes b$, where $A$ is an $(m+1) \times (n+1)$ matrix with $A_{ij} = ni-mj$ and $b$ is an $r \times 1$ matrix with all $1$s. Noticing that it has such a simple structure, you should be able to get a lot of information about it. For example, you can prove your claim that it's rank is always 2. –  polkjh Jan 15 '13 at 16:52

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