Euclidean Norm (L₂) Calculator

About Euclidean Norm (L₂)

The Euclidean norm, also known as L₂ norm or vector length, measures the straight-line distance from the origin to a point in space. It's the most commonly used vector norm and corresponds to our intuitive notion of distance.

Formula:

For a vector \(\vector{x} = (x_1, x_2, \ldots, x_n)\):
\[ \norm{\vector{x}}_2 = \sqrt{\sum_{i=1}^n x_i^2} = \sqrt{x_1^2 + x_2^2 + \cdots + x_n^2} \]

Properties:

Non-negativity: \(\norm{\vector{x}}_2 \geq 0\)
Positive definiteness: \(\norm{\vector{x}}_2 = 0\) if and only if \(\vector{x} = \vector{0}\)
Homogeneity: \(\norm{\alpha\vector{x}}_2 = \abs{\alpha}\norm{\vector{x}}_2\)
Triangle inequality: \(\norm{\vector{x} + \vector{y}}_2 \leq \norm{\vector{x}}_2 + \norm{\vector{y}}_2\)
Pythagorean theorem in \(n\) dimensions

Applications:

Distance measurement in physical space
Least squares optimization
Signal energy calculation
Principal Component Analysis (PCA)
Neural network weight regularization

Example:

For the vector \(\vector{x} = (3, 4)\):
\[ \norm{\vector{x}}_2 = \sqrt{3^2 + 4^2} = \sqrt{9 + 16} = \sqrt{25} = 5 \] This is the familiar 3-4-5 right triangle!

Connection to Inner Product:

The Euclidean norm is related to the inner product:
\[ \norm{\vector{x}}_2 = \sqrt{\langle \vector{x}, \vector{x} \rangle} \]