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Possibly a stupid question, but are Toeplitz matrices always square?

Wikipedia seems to suggest so, as the Toeplitz page says

"... Any $n \times n$ matrix $A$ of the form..."

But Matlab suggests otherwise:

>> toeplitz([0,1,2,3],[0,4,5])

ans =

     0     4     5
     1     0     4
     2     1     0
     3     2     1

Is Matlab just being helpful here or is that a valid Toeplitz matrix?

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    $\begingroup$ The Wikipedia article doesn't define a Toeplitz matrix as square. It says that "any $n\times n$ matrix of the form [...] is Toeplitz", but doesn't say that any Toeplitz matrix is $n\times n$. The only definition given is that a Toeplitz matrix is one in which "every descending diagonal from left-to-right is constant", which includes many non-square matrices. Additionally, see Wolfram Alpha $\endgroup$
    – Chris Taylor
    Jan 30, 2012 at 13:14
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    $\begingroup$ I'm with Chris; the only defining characteristic of Toeplitz matrices is that they have constant diagonals (and similarly, Hankel matrices are matrices with constant antidiagonals). Nothing about being square there. $\endgroup$
    – J. M. ain't a mathematician
    Jan 30, 2012 at 13:56

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It's not a stupid question, so much as it's hard to answer. In my search for definitions, I have found some that require the matrix to be square and I have found some that do not mention that the matrix is square, but the only examples given are square. I imagine the answer is that the original definition was for square matrices but perhaps some people started using the term to describe generalizations. Or, perhaps non-square matrices of this type aren't that interesting.

The definition given in Matrix Analysis by Horn and Johnson is:

A matrix $A = [a_{ij}] \in M_{n+1}$ of the form

$$A = \begin{bmatrix} a_0 & a_1 & a_2 & \cdots & a_n \\ a_{-1} & a_0 & a_1 & \cdots & a_{n-1} \\ a_{-2} & a_{-1} & a_0 & \cdots & \vdots \\ \vdots & \vdots & \ddots & \ddots & a_1 \\ a_{-n} & a_{-n+1} & \cdots & a_{-1} & a_0 \end{bmatrix}$$

is called a Toeplitz matrix.

As far as J.M.'s comment, right after this, it defines Hankel matrices and requires that they are square as well. In fact, not long after, it says "Since ... Hankel matrices are symmetric...", which requires that Hankel matrices are square.

This set of notes I found online, which contains 98 pages about Toeplitz matrices, require them to be square by definition.

On the other hand, Matrix Analysis and Applied Linear Algebra by Meyer gives the definition:

Toeplitz matrices have constant entries on each diagonal parallel to the main diagonal.

So, this doesn't require the matrix to be square, but then they give a few examples, all of which are square.

By the way, just because MatLab or Mathematica allow something doesn't mean the definition actually allows those things, especially if it doesn't take any more work to program something.

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  • $\begingroup$ Do you a reference to solve Linear System with a non rectangular Toeplitz Matrix? $\endgroup$
    – Royi
    Jan 11, 2020 at 17:05

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