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Algebra was years ago and I cannot remember what kind of operation the | means. For example:

Matrices

Where M is a 3x3 matrix and C a column-vector.

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    $\begingroup$ I would guess is the concatenation of $M$ with (minus)the resulting vector $MC$. So $P$ is $3\times 4$. $\endgroup$
    – Zubzub
    Sep 21, 2017 at 12:26
  • $\begingroup$ But what is the concatenation? As I said, my matrices knowledge is long gone :X $\endgroup$
    – Gonçalo
    Sep 21, 2017 at 12:35
  • $\begingroup$ @GonacFaria concatenation means adjoining them one after the other. For example concatenating the strings "far" and "out" would give me "farout" or "outfar" depending on the order of concatenation. $\endgroup$ Sep 21, 2017 at 13:03

2 Answers 2

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Say we have $$ M = \begin{pmatrix} a & b & c \\ d & e & f \\ g & h & i \end{pmatrix} , \hspace{1cm} C = \begin{pmatrix} x \\ y \\ z \end{pmatrix} $$ So $$ -MC = \begin{pmatrix} -ax - by - cz \\ -dx - ey - fz \\ -gx - hy - iz \end{pmatrix} $$ Then $$ P = [M | -MC] = \begin{pmatrix} a & b & c & (-ax - by - cz)\\ d & e & f & (-dx - ey - fz) \\ g & h & i & (-gx - hy - iz) \end{pmatrix} \in \mathbb{R}^{3 \times 4} $$

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It is the notation for an augmented matrix. Namely, it is a 3 by 4 matrix where the last column is the column vector $\mathbf{-MC}$.

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