Vector Normalization - Euclidean column-norm scaling (L2 normalisation)

Overview

After vector normalisation, each column has unit Euclidean norm (Σ_i r_ij² = 1). The normalised values preserve the ratio structure of the original data. Cost/benefit direction is NOT applied during this step - it must be handled downstream (e.g. by selecting A⁺/A⁻ in TOPSIS).

Output

normalized matrix, higher is better

Data

Crisp, complete numeric matrix

Size

2+ alternatives, 3-12 criteria works best

Used for

Preprocessing

How it works

1. 1

   For each criterion j compute the column Euclidean norm ||x_j||₂ = √(Σ_i x_ij²).

   $$\Vert {\mathbf{x}}_j{\Vert }_2=\sqrt{\sum _{i=1}^m{x}_{ij}^2}$$

   Hwang & Yoon 1981, p.130 Eq.(4.1) (pending PDF page verification)

2. 2

   Divide each entry by its column norm: r_ij = x_ij / ||x_j||₂. Direction awareness (benefit/cost) is handled by the downstream ranking method, not during normalisation.

   $$r_{ij}=\frac{x_{ij}}{\sqrt{{\sum }_{k=1}^m{x}_{kj}^2}}$$

   Hwang & Yoon 1981, p.130 Eq.(4.1) (pending PDF page verification)

Edge cases and pitfalls

• All-zero column: L2 norm = 0, division undefined. Ensure no criterion has all-zero performance values.

• Negative values: r_ij can be negative (L2 norm still well-defined). Downstream methods may require non-negative inputs - check compatibility.

Works with

How to cite