# Vector notation for vector with single observation removed

Given a vector $x \in \mathbb{R}^n$, what would be a good notation for describing the vector $x$ with the $i$-th element removed. That is, if one wishes to remove index $j$, then the elements of our new vector $x' \in \mathbb{R}^{n-1}$ will be:

$$x'_{i = [1,n-1)} = \begin{cases} x_i & \text{if } i < j\\ x_{i+1} & \text{if } i \geq j\\ \end{cases}$$

I have considered something like $x_{-j}$, but not sure if that is acceptable, or if there is already a common notation for this.

• The passage from one to the other is a projection, you can just denote it as such. Call $\pi_i$ the projection $\pi_i:\mathbb{R}^n\to\mathbb{R}^{n-1}$ that drops the $i$-th component. Then your vector is $\pi_i(x)$. – Hellen Jul 21 '17 at 12:12
• @Hellen Nice, I like that. You should make that a solution! – Patrick Jul 21 '17 at 15:10

For a given vector $x∈ℝ^n$ we define
$$x^i:= \begin{pmatrix}x_1 \\ \vdots \\ x_{i-1} \\ x_{i+1} \\ \vdots \\x_n \end{pmatrix}∈ℝ^{n-1}$$ which has the $i$-th element removed.
I don't think there is a standard notation, but I would define a new vector: $$a := x \setminus \{x_j\}$$ This notation considers the vector as a set of components; the backslash represents the minus operator used in set theory.