Confusion about the notation of transpose

[alyafey22]

Define the following

$$Z=
\begin{pmatrix}
0 & A \\
B^T & T
\end{pmatrix}$$

where we define $A$ and $B$ as $r \times m $ matrices and $T$ is an $m \times m$ matrix with nonzero distinct indeterminates at the diagonal, that is, $T_{i,i} = t_i$.

What is the meaning of $B^T$ ?

 

[Opalg]

Define the following

$$Z=
\begin{pmatrix}
0 & A \\
B^T & T
\end{pmatrix}$$

where we define $A$ and $B$ as $m \times r $ matrices and $T$ is is an $m \times m$ matrix with nonzero distinct indeterminates at the diagonal, that is, $T_{i,i} = t_i$.

What is the meaning of $B^T$ ?

$B^{\textsf{T}}$ denotes the transpose of $B$. So if $B$ is an $r\times m$ matrix with entries $b_{ij}$ then $B^{\textsf{T}}$ is an $m\times r$ matrix whose $(i,j)$-entry is $b_{ji}.$

Notice that $A$ and $B$ should be $r\times m$ matrices, not $m\times r$. That way, $A$ has the same number of columns as $T$, and $B^{\textsf{T}}$ has the same number of rows as $T$. To complete the partitioned matrix, the $0$ in the top left corner of $Z$ represents an $r\times r$ matrix consisting of all zeros.

 

[alyafey22]

Notice that $A$ and $B$ should be $r\times m$ matrices, not $m\times r$. That way, $A$ has the same number of columns as $T$, and $B^{\textsf{T}}$ has the same number of rows as $T$

You are correct , that was a typo. But my confusion is for using $T$ as a matrix and as a transpose. I know that $B^T$ normally defines the transpose of a matrix but is $T$ in the first place a matrix ? or just a notation ?

 

[Opalg]

But my confusion is for using $T$ as a matrix and as a transpose. I know that $B^T$ normally defines the transpose of a matrix but is $T$ in the first place a matrix ? or just a notation ?

I hadn't even noticed that! It's just a case of bad notation. The $\textsf{T}$ that denotes the transpose is nothing to do with the $T$ that denotes the matrix. I'm glad that in my previous comment I automatically used different typefaces for them.

 

[alyafey22]

I found that construction in a published paper about Linear matroid parity. I am surprised that the authors are using such a bad notation :/