Norm

Shipping

matrix_norm

Selectable vector/matrix norm (Euclidean, L1, L∞, Lᵖ, L0, spectral, nuclear, operator)

Signature

Inputs

aSignal|Arrayrequired— A Vector, Matrix, or Tensor. Entrywise norms accept any rank; matrix norms require rank 2.

Outputs

normScalar— The selected norm value; carries the input's unit (except the dimensionless L0 count).

Parameters

Key	Type	Default	Notes
`norm`	enum	euclidean	one of: euclidean, l1, linf, lp, l0, spectral, nuclear, operator_1, operator_inf
`p`	float	2	Exponent for the general Lᵖ norm (Σ\|xᵢ\|ᵖ)^(1/p). p ≥ 1 is a true norm; 0 < p < 1 a quasi-norm. Shown only when Norm = Lᵖ.

Description

Norm is the one rank-polymorphic member of the linear-algebra family: a single ‖·‖ output whose value is chosen by the norm dropdown — and only the selected norm is computed, so you never pay for the ones you didn't ask for.

The entrywise norms — Euclidean/Frobenius ( $\sum x^{2}$ ), L1 ( $\sum ∣ x ∣$ ), L∞ ( $max ∣ x ∣$ ), the general Lᵖ ( $(\sum ∣ x ∣^{p})^{1/ p}$ , with the exponent p), and L0 (count of nonzeros) — flatten the array and apply at any rank: a vector, matrix, or tensor. The matrix-only norms require a 2-D input: spectral (largest singular value $σ_{m a x}$ ), nuclear (trace norm $\sum σ_{i}$ ), and the induced operator norms — operator-1 (max column sum) and operator-∞ (max row sum). A matrix-only norm applied to a vector raises a clear backend error.

Because a norm is degree-1 homogeneous ( $∥ αx ∥ = ∣ α ∣ ∥ x ∥$ ), the output inherits the input's unit — with one exception: the L0 count is a dimensionless sparsity measure and carries no unit.

Mathematics

∥ x ∥_{2} = \sum_{i} x_{i}^{2}, ∥ x ∥_{1} = \sum_{i} ∣ x_{i} ∣, ∥ x ∥_{\infty} = max_{i} ∣ x_{i} ∣

∥ x ∥_{p} = (\sum_{i} ∣ x_{i} ∣^{p})^{1/ p}, ∥ x ∥_{0} = # {i : x_{i} \neq = 0}

∥ A ∥_{2} = σ_{m a x}, ∥ A ∥_{*} = \sum_{i} σ_{i}, ∥ A ∥_{1} = max_{j} \sum_{i} ∣ A_{ij} ∣

Examples

Euclidean norm of a vector

With norm = euclidean (the default) and input $[3, - 4]$ in metres, the output is $3^{2} + 4^{2} = 5$ — and it keeps the m unit. Switch the dropdown to l1 for $7$ , linf for $4$ , or l0 for $2$ (both entries nonzero, dimensionless).

Matrix norms on a 2×2

For $A = [1002]$ : euclidean (Frobenius) is $5$ , operator_1 and operator_inf are $2$ , spectral is $2$ ( $σ_{m a x}$ ), and nuclear is $3$ ( $σ_{1} + σ_{2}$ ). Selecting a matrix-only norm on a vector instead raises a 'needs a 2-D matrix' error.

Applications

Measuring signal energy (L2), total variation (L1), or peak magnitude (L∞) of a vector.
Regularization / sparsity metrics: L1, L0, and nuclear norms for compressed sensing and low-rank recovery.
Quantifying matrix magnitude and gain via spectral/operator norms in stability analysis.
Frobenius norm as an overall error measure between matrices.

Neat

It computes strictly the one norm the dropdown selects — a matrix never runs the extra operator/SVD passes unless you actually pick a spectral/nuclear/operator norm.

Entrywise norms are rank-agnostic (they flatten the buffer), so the same node handles vectors, matrices, and tensors, while matrix-only norms self-validate rank 2 with a clear error.

Unit handling encodes the math: norms are degree-1 homogeneous so they keep the input's unit, but the L0 nonzero-count is treated as a dimensionless sparsity measure.

Known issues

Selecting spectral, nuclear, or an operator norm on a non-2-D input errors (by design) — those norms are matrix-only.

The Lᵖ norm with $0<p<1$ is a quasi-norm (violates the triangle inequality), so treat sub-1 exponents as a sparsity-style measure, not a true norm.