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I am searching for an $m \times n$ matrix that has the following sparsity pattern

\begin{bmatrix} a_1 & a_2 & a_3 & 0 & 0 & 0 & 0 &0 & 0 & 0 \\ a_4 & a_5 & a_6 & a_7 & a_8 & a_9 & 0 & 0 & 0 & 0 \\ \vdots & \vdots & \vdots & \vdots & \vdots & \vdots & \vdots & \vdots & \vdots & \vdots\\ a_i & a_{i+1} & a_{i+2} & \dots & \dots & \dots & \dots & a_{n-2} & a_{n-1} & a_n \end{bmatrix}

The number of zeros on each row can be known ahead of time, and are decreasing until the final row which is all non-zero. In general, $n$ is much larger than $m$ (it's a wide/fat matrix).

Do these types of matrices have names? Do they live in some set in which I can constrain my optimization problem? If not, how can I encode this structure within an optimization problem?

I think that the fact that this matrix has so many zeros, and that the number of the zeros on each row are known in advance should make the problem easier.

EDIT: To clarify, a matrix of this structure is my optimization variable.

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    Block-lower triangular? (Notice that high-performance LP solvers will exploit sparsity in the technological coefficient matrix $A$). – 2017-01-24
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    If you put the columns in reversed order, the matrix is in row echelon form. Your question is not clear as to how this matrix appears in your optimization problem; is it a variable or a coefficient matrix? – 2017-01-24
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    @ErwinKalvelagen can you explain what you mean by "technological coefficient"? – 2017-01-24
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    @LinAlg thanks for you response, and sorry I wasn't clear - I edited the question – 2017-01-24
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    And how does this matrix occur in the optimization problem? – 2017-01-24
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    It is something like $\underset{M}{min} \;\; M\cdot\vec{v}$ – 2017-01-24
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    @RodrigodeAzevedo sorry Im not understanding - can't it be something $\underset{M}{\min} \;\; \textbf{Tr}(M) - \log\det M - \lvert\lvert X\rvert\rvert_F$ – 2017-01-24
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    I edited the above - Frobenius for example? I got this problem from a textbook, though it doesnt mention the unique structure of $M$ in that textbook - that comes from my personal research – 2017-01-24
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    Why not just include $(m - 1) n$ equality constraints of the form $x_{ij} = 0$? – 2017-01-24
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    Is that how it is best handled? thats what I was wondering - if this is a special matrix with a unique structure, or if I just have to include those constraints. – 2017-01-24
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    The number of zeros are combinatorial, related to a weak integer composition – 2017-01-24
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    I would just use $x_{ij} = 0$. Write a script that generates the data of the problem. Then let a solver handle it. – 2017-01-24
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    @LinAlg It is a coefficient matrix over some space of basis functions - but what I am trying to find are the coefficients over that space – 2017-01-24

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