What Is a Norm in Linear Algebra?

A norm is a function that measures the size or length of a vector, matrix, or more general mathematical object. In elementary geometry, the length of a vector is usually its Euclidean length. Linear algebra uses several different norms because different applications—such as optimization, numerical computation, and functional analysis—benefit from different notions of size.

The most common vector norm is the Euclidean norm, also called the 2-norm. However, the 1-norm and infinity-norm are also standard and often easier to compute or analyze.

The definition of a vector norm

Let $V$ be a real or complex vector space. A function

$$\|\cdot\|:V\to [0,\infty)$$

is a norm if, for all vectors $x,y\in V$ and all scalars $c$, it satisfies:

1. Positive definiteness: $$\|x\|\ge 0,\qquad \|x\|=0 \iff x=0.$$
2. Absolute homogeneity: $$\|cx\|=|c|\,\|x\|.$$
3. Triangle inequality: $$\|x+y\|\le \|x\|+\|y\|.$$

The first property says that a norm is never negative and that only the zero vector has size zero. The second says that scaling a vector scales its length by the absolute value of the scalar. The triangle inequality expresses the idea that going directly from one point to another is no longer than taking a two-segment route.

A vector space equipped with a norm is called a normed vector space.

Common norms on $\mathbb{R}^n$ and $\mathbb{C}^n$

For a vector $x=(x_1,x_2,\ldots,x_n)$, the most frequently used norms are:

The 1-norm

$$\|x\|_1=|x_1|+|x_2|+\cdots+|x_n|.$$

This is the sum of the absolute values of the coordinates. It is sometimes called the taxicab norm because, in two dimensions, it measures movement along a rectangular grid.

The 2-norm

$$\|x\|_2=\sqrt{|x_1|^2+|x_2|^2+\cdots+|x_n|^2}.$$

For real vectors, this is the ordinary Euclidean length. In a real inner-product space, it can be written as

$$\|x\|_2=\sqrt{x^{\mathsf T}x}.$$

For complex vectors, the corresponding expression uses the conjugate transpose:

$$\|x\|_2=\sqrt{x^*x}.$$

The infinity-norm

$$\|x\|_\infty=\max_{1\le i\le n}|x_i|.$$

This norm takes the largest absolute coordinate. It is also called the maximum norm or supremum norm in finite-dimensional settings.

More generally, for $1\le p<\infty$, the $p$-norm is

$$\|x\|_p=\left(\sum_{i=1}^n |x_i|^p\right)^{1/p}.$$

The 1-norm and 2-norm are special cases. The infinity-norm is obtained as the limiting case as $p$ tends to infinity.

Worked example

Take

$$x=(-2,3,-6).$$

Its 1-norm is

$$\|x\|_1=|-2|+|3|+|-6|=2+3+6=11.$$

Its 2-norm is

$$\begin{aligned} \|x\|_2 &=\sqrt{(-2)^2+3^2+(-6)^2}\\ &=\sqrt{4+9+36}\\ &=\sqrt{49}=7. \end{aligned}$$

Its infinity-norm is

$$\|x\|_\infty=\max\{2,3,6\}=6.$$

Thus the same vector has three different lengths:

$$\|x\|_1=11,\qquad \|x\|_2=7,\qquad \|x\|_\infty=6.$$

There is no contradiction. The word “norm” does not identify one unique formula; it identifies any function satisfying the norm axioms.

Matrix norms

Matrices can also be assigned norms. Two important types should be distinguished.

The Frobenius norm

For a matrix $A=(a_{ij})\in\mathbb{R}^{m\times n}$, the Frobenius norm is

$$\|A\|_F=\left(\sum_{i=1}^m\sum_{j=1}^n |a_{ij}|^2\right)^{1/2}.$$

It treats the matrix as one long list of entries. Equivalently,

$$\|A\|_F=\sqrt{\operatorname{tr}(A^*A)}.$$

For example, let

$$A=\begin{pmatrix}1&-2\\3&4\end{pmatrix}.$$

Then

$$\|A\|_F=\sqrt{1^2+(-2)^2+3^2+4^2}=\sqrt{30}.$$

Induced, or operator, matrix norms

A matrix can also be viewed as a linear transformation. Given a vector norm, its induced matrix norm is

$$\|A\|=\max_{x\ne 0}\frac{\|Ax\|}{\|x\|}.$$

This measures the largest factor by which $A$ can stretch a nonzero vector. Equivalently,

$$\|A\|=\max_{\|x\|=1}\|Ax\|.$$

The induced norms associated with the vector 1-, 2-, and infinity-norms have useful formulas:

$$\|A\|_1=\max_j\sum_i |a_{ij}|,$$

so the matrix 1-norm is the largest absolute column sum;

$$\|A\|_\infty=\max_i\sum_j |a_{ij}|,$$

so the matrix infinity-norm is the largest absolute row sum; and

$$\|A\|_2=\sqrt{\lambda_{\max}(A^*A)},$$

where $\lambda_{\max}$ is the largest eigenvalue.

For the example above,

$$\|A\|_1=\max\{1+3,2+4\}=6,$$

and

$$\|A\|_\infty=\max\{1+2,3+4\}=7.$$

These are not the Frobenius norm, which is $\sqrt{30}\approx 5.477$.

Why the choice of norm matters

In finite-dimensional spaces, all norms generate the same notion of convergence: a sequence converges under one norm if and only if it converges under any other norm. Nevertheless, the numerical values can differ substantially, and some norms are better suited to particular tasks.

The 1-norm promotes sparsity in many optimization problems. The 2-norm is natural for Euclidean geometry and least-squares calculations. The infinity-norm is useful when the largest coordinate error must be controlled.

For every $x\in\mathbb{R}^n$, these common norms satisfy inequalities such as

$$\|x\|_\infty\le \|x\|_2\le \|x\|_1,$$

and

$$\|x\|_1\le \sqrt{n}\,\|x\|_2,\qquad \|x\|_2\le \sqrt{n}\,\|x\|_\infty.$$

These inequalities explain why changing norms changes measurements but not the basic topology of finite-dimensional space.

Common mistakes

• Assuming “the norm” is unique: Always check whether the problem means the 1-, 2-, infinity-, Frobenius, or an induced matrix norm.
• Forgetting absolute values: Norm formulas use $|x_i|$, not $x_i$. Negative coordinates still contribute positively.
• Confusing a matrix norm with a vector norm: The Frobenius norm sums all matrix entries, while induced norms measure how a matrix acts on vectors.
• Using a square instead of a square root: The Euclidean norm is $\sqrt{x_1^2+\cdots+x_n^2}$; the expression inside the square root is the squared norm.
• Calling a seminorm a norm: A seminorm may assign value zero to a nonzero vector, so it fails positive definiteness.