What Is an Example of a Nonstandard Inner Product?

A simple example of an inner product that is not the usual dot product is a weighted dot product on $\mathbb{R}^2$. It gives different importance to the two coordinates while still satisfying all the inner-product axioms.

The phrase “non-Euclidean inner product” can be misleading. In many linear algebra contexts, it means an inner product other than the standard dot product in the coordinates being used—not a geometry that is non-Euclidean in the sense of curved space.

A weighted inner product on $\mathbb{R}^2$

For vectors $x=(x_1,x_2)$ and $y=(y_1,y_2)$, define

$$\langle x,y\rangle=2x_1y_1+3x_2y_2.$$

This differs from the standard dot product, $x_1y_1+x_2y_2$, because the first coordinate is weighted by $2$ and the second by $3$.

To check that it is an inner product over the real numbers, verify the defining properties:

• Linearity in each argument: The expression is linear in the coordinates of $x$ and also in the coordinates of $y$.
• Symmetry: $\langle x,y\rangle=2x_1y_1+3x_2y_2=\langle y,x\rangle$.
• Positive definiteness:

$$\langle x,x\rangle=2x_1^2+3x_2^2\geq 0,$$

and this is zero only when $x_1=x_2=0$. Thus $\langle x,x\rangle>0$ for every nonzero vector $x$.

All three properties hold, so this is a valid inner product.

A worked example: lengths and orthogonality

Take $u=(1,1)$ and $v=(3,-2)$. Their inner product is

$$\langle u,v\rangle=2(1)(3)+3(1)(-2)=6-6=0.$$

Therefore, $u$ and $v$ are orthogonal under this weighted inner product. Under the standard dot product, however,

$$u\cdot v=(1)(3)+(1)(-2)=1,$$

so they are not orthogonal. The choice of inner product can change which vectors count as perpendicular.

It also changes lengths. The length induced by an inner product is $\|x\|=\sqrt{\langle x,x\rangle}$. For the coordinate vectors $e_1=(1,0)$ and $e_2=(0,1)$, this gives

$$\|e_1\|=\sqrt{2},\qquad \|e_2\|=\sqrt{3}.$$

In the standard dot product, both coordinate vectors have length $1$. The unit vectors for the weighted inner product satisfy $2x_1^2+3x_2^2=1$, which describes an ellipse in the usual coordinate plane rather than the standard unit circle.

The matrix form

The same inner product can be written as

$$\langle x,y\rangle=x^{\mathsf T}Ay, \qquad A=\begin{pmatrix}2&0\\0&3\end{pmatrix}.$$

For a real matrix $A$, this formula defines an inner product on $\mathbb{R}^n$ when $A$ is symmetric positive definite. Symmetry makes the inner product symmetric, and positive definiteness ensures $x^{\mathsf T}Ax>0$ for every nonzero $x$. Here, $A$ is diagonal with positive diagonal entries, so it is positive definite.

More generally, any choice of positive weights $w_1,\ldots,w_n$ gives an inner product

$$\langle x,y\rangle=\sum_{i=1}^n w_i x_i y_i, \qquad w_i>0.$$

If a weight is zero or negative, positive definiteness fails: a nonzero vector can have squared length zero or less than zero.

What “non-Euclidean” means here

A finite-dimensional real inner-product space is isometric to $\mathbb{R}^n$ with its standard dot product: an appropriate change of coordinates preserves inner products. So the weighted example is not a fundamentally different kind of finite-dimensional geometry. It is nonstandard in the given coordinates, and the changed lengths and angles are useful in applications such as least squares with weighted errors.

Common mistakes

• Changing the formula without checking positivity: A symmetric bilinear form is not automatically an inner product. For example, $x_1y_1-x_2y_2$ is symmetric and bilinear, but its value at $x=y=(0,1)$ is $-1$.
• Calling every different formula invalid: An inner product need not be the standard dot product. Positive weights, or more generally a symmetric positive-definite matrix, give legitimate alternatives.
• Treating “non-Euclidean” as a precise universal term: In this setting it usually means “not the standard coordinate dot product.” If the term is intended to mean curved or non-Euclidean geometry, a different question is being asked.