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A partial matrix over a field $\mathbb{F}$ is a matrix whose entries are either an element of $\mathbb{F}$ or an indeterminate and with each indeterminate only appearing once. A completion is an assignment of values in $\mathbb{F}$ to all indeterminates. Given a partial matrix, through elementary row operations and column permutation it can be decomposed into a block matrix of the form $\left[\begin{smallmatrix}{\bf W} & * & * \\ 0 & {\bf S} & * \\ 0 & 0 & {\bf T} \end{smallmatrix}\right]$ where ${\bf W}$ is wide (has more columns than rows), ${\bf S}$ is square, ${\bf T}$ is tall (has more rows than columns), and these three blocks have at least one completion with full rank. And importantly, each one of the blocks ${\bf W}$, ${\bf S}$ and ${\bf T}$ is unique up to elementary row operations and column permutation whenever ${\bf S}$ is required to be as large as possible. When this is the case $\left[\begin{smallmatrix}{\bf W} & * & * \\ 0 & {\bf S} & * \\ 0 & 0 & {\bf T} \end{smallmatrix}\right]$ will be called a WST-decomposition. With this decomposition it is trivial to compute maximum rank of a completion of the original partial matrix: $\#\mbox{rows}({\bf W})+\#\mbox{rows}({\bf S})+\#\mbox{cols}({\bf T})$. In fact we introduce the WST-decomposition for a broader class of matrices: the ACI-matrices.

We show for $n,k\geq1$, and an $n$-dimensional complex vector space $V$ that if an element $A\in\text{End}(V)[[z]]$ has constant term similar to a Jordan block, then there exists a polynomial gauge transformation $g$ such that the first $k$ coefficients of $gAg^{-1}$ have a controlled normal form. Furthermore, we show that this normal form is unique by demonstrating explicit relationships between the first $nk$ coefficients of the Puiseux series expansion of the eigenvalues of $A$ and the entries of the first $k$ coefficients of $gAg^{-1}$., Comment: 13 pages, to appear in Involve: A Journal of Mathematics

The columns of a $m\times n$ ACI-matrix over a field $\mathbb{F}$ are independent affine subspaces of $\mathbb{F}^m$. An ACI-matrix has constant rank $\rho$ if all its completions have rank $\rho$. Huang and Zhan (2011) characterized the $m\times n$ ACI-matrices of constant rank when $|\mathbb{F}|\geq \min\{m,n+1\}$. We complete their result characterizing the $m\times n$ ACI-matrices of constant rank over arbitrary fields. Quinlan and McTigue (2014) proved that every partial matrix of constant rank $\rho$ has a $\rho\times \rho$ submatrix of constant rank $\rho$ if and only $|\mathbb{F}|\geq \rho$. We obtain an analogous result for ACI-matrices over arbitrary fields by introducing the concept of complete irreducibility., Comment: 20 pages

Let [math] be a unital ring and let [math] be any upper triangular matrix of trace zero. Then there exist matrices [math] and [math] in [math] such that [math] .

The resultant of two tropical polynomials satisfies the similar properties to the resultant of two polynomials over a field.

The paper studies algebraic strong shift equivalence of matrices over $n$-variable polynomial rings over a principal ideal domain $D$($n\leq 2$). It is proved that in the case $n=1$, every non-zero matrix over $D[x]$ has a full rank factorization and every non-nilpotent matrix over $D[x]$ is algebraically strong shift equivalent to a nonsingular matrix. In the case $n=2$, an example of non-nilpotent matrix over $\mathbb{R}[x,y,z]=\mathbb{R}[x][y,z]$, which can not be algebraically shift equivalent to a nonsingular matrix, is given., Comment: 8 pages