How do you truncate columns from a matrix with algebraic operations?

Question

I have a matrix of dimension $n \times m$ and want to truncate a column left or right side, to get the remaining $n \times m-1$ matrix (one column removed). How can I do this using standard algebraic notation?

Answer 1

Let $\mathrm{M}$ be the $n \times m$ matrix and $\mathrm{N}$ be the truncated $n \times m-1$ matrix. Then define:

$\mathrm{R} = \begin{bmatrix} \mathrm{I}_{m-1} \\ 0_{1,m-1} \end{bmatrix} $ as an $m \times m-1$ right truncating matrix

$\mathrm{L} = \begin{bmatrix} 0_{1,m-1}\\ \mathrm{I}_{m-1} \end{bmatrix} $ as an $m \times m-1$ left truncating matrix

then truncate the right column as follows:

$\mathrm{N}$ = $\mathrm{M}\mathrm{R}$

and truncate the left column as follows:

$\mathrm{N}$ = $\mathrm{M}\mathrm{L}$

Uses only standard matrix multiplication and nomenclature commonly found and well documented as here:

which uses only standard notation from:

• https://en.wikipedia.org/wiki/Identity_matrix
• https://en.wikipedia.org/wiki/Zero_matrix
• https://en.wikipedia.org/wiki/Block_matrix