Notation for replacing a matrix column with a vector

Question

Let $A$ be an $n\times n$ matrix. Let $v$ be an $n\times 1$ matrix. Is there a notation to signify replacing the $j$-th column of $A$ with $v$? If not, what is the accepted way to denote this?

Answer 1

I am not aware of a universal notation for this, however, since I work a lot with MATLAB, I take the code notation used to represent the matrix you asked for, that is $$ [A(:,1:j-1), v, A(:,j+1:n)] $$ I am aware this is not usual at all but it works fine for me. My numerical linear algebra teacher used to represent that matrix by $$ A_{\left[\vec{v}, j \right]} $$ But again not as a usual notation. Hope it helps.

Answer 2

Allow me to make a suggestion that is:

• less compact, and
• completely non-standard.

Haha. But on the plus side, it may actually help you write better, cleaner proofs.

A bit of background: recently I've been pondering the "else" monoid, defined here. I'll just quickly redefine it using the notation I currently prefer. Given a set $X$, we write $X_\bot$ for the result of adjoining an extra element $\bot$ to $X$. So: $$X_\bot = X \cup \{\bot\}.$$ Now define a binary operation $\triangledown$ on $X_\bot$ as follows: given $x',x \in X_\bot$, if $x'$ is distinct from $\bot$, then $x' \mathop{\triangledown} x$ equals $x'$. Otherwise, $x' \mathop{\triangledown} x$ equals $x$.

For example:

• $17 \mathop{\triangledown} 3 = 17$,
• $14 \mathop{\triangledown} 3 = 14$,
• $\bot \mathop{\triangledown} 3 = 3$

I call $\mathop{\triangledown}$ the "else" operation, and I read $x' \mathop{\triangledown} x$ as "$x'$ else $x$."

Here's the cool thing: by computing entrywise, the else operation can be used to replace columns. For example:

$$\begin{bmatrix} \bot & a & \bot \\ \bot & b & \bot \\ \bot & c & \bot \\ \end{bmatrix} \mathop{\triangledown} \begin{bmatrix} x_{11} & x_{12} & x_{13} \\ x_{21} & x_{22} & x_{23} \\ x_{31} & x_{32} & x_{33} \\ \end{bmatrix} = \begin{bmatrix} \bot \mathop{\triangledown} x_{11} & a \mathop{\triangledown} x_{12} & \bot \mathop{\triangledown} x_{13} \\ \bot \mathop{\triangledown} x_{21} & b \mathop{\triangledown} x_{22} & \bot \mathop{\triangledown} x_{23} \\ \bot \mathop{\triangledown} x_{31} & c \mathop{\triangledown} x_{32} & \bot \mathop{\triangledown} x_{33} \end{bmatrix} = \begin{bmatrix} x_{11} & a & x_{13} \\ x_{21} & b & x_{23} \\ x_{31} & c & x_{33} \\ \end{bmatrix}$$

Therefore, my suggestion is:

1. Firstly, define that if $v$ is a column vector with $R$ rows, then $\mathrm{col}_{c,C}(v)$ is the $R \times C$ matrix whose $c$th column is $v$.

2. It then follows that if $A$ is an $R \times C$ matrix and $v$ is a column vector with $R$ rows, then the result of replacing the $c$th column of $A$ with the vector $v$ can be denoted:

$$\mathrm{col}_{c,C}(v) \mathop{\triangledown} A$$

Okay, its not very compact, but you may actually find the above notation more useful than a more compact "black box" style of notation. For something shorter, you may wish to define:

$$\mathrm{col}_{c}(v,A) = \mathrm{col}_{c,C}(v) \mathop{\triangledown} A$$