Weighted Dot Products With Positive Weights

Why Weight the Coordinates?

The standard dot product gives each coordinate an equal contribution. In some settings, however, one coordinate should contribute more to a measured length or interaction than another. Positive weights provide a direct way to adjust those contributions: a large weight makes a coordinate count more, while a small positive weight makes it count less. The key condition is that every weight must be strictly positive. This is what ensures the resulting product still satisfies the real inner product axioms.

We work over \(\mathbb{R}\). Recall that a real inner product on \(\mathbb{R}^n\) is a function \(\langle\cdot,\cdot\rangle:\mathbb{R}^n\times\mathbb{R}^n\to\mathbb{R}\) that is linear in each argument, symmetric, and positive definite. Positive definiteness means \(\langle\mathbf{x},\mathbf{x}\rangle>0\) whenever \(\mathbf{x}\ne\mathbf{0}\), and \(\langle\mathbf{0},\mathbf{0}\rangle=0\).

The subscript \(w\) indicates that this product need not be the standard dot product. In particular, the weight on coordinate \(i\) multiplies that coordinate's contribution \(x_i y_i\). The corresponding squared length is \(\sum_{i=1}^n w_i x_i^2\), so each coordinate contributes a nonnegative amount, and a nonzero coordinate contributes a strictly positive amount.

Why Strictly Positive Weights Give an Inner Product

Proof. First suppose every \(w_i>0\). For any \(\mathbf{x},\mathbf{y}\in\mathbb{R}^n\), commutativity of real multiplication gives \(\langle\mathbf{x},\mathbf{y}\rangle_w=\sum_{i=1}^n w_i x_i y_i=\sum_{i=1}^n w_i y_i x_i=\langle\mathbf{y},\mathbf{x}\rangle_w\), so the product is symmetric.

For any \(\mathbf{x},\mathbf{y},\mathbf{z}\in\mathbb{R}^n\) and \(a,b\in\mathbb{R}\), the \(i\)th coordinate of \(a\mathbf{x}+b\mathbf{y}\) is \(ax_i+by_i\). Therefore \(\langle a\mathbf{x}+b\mathbf{y},\mathbf{z}\rangle_w=\sum_{i=1}^n w_i(ax_i+by_i)z_i=a\sum_{i=1}^n w_i x_i z_i+b\sum_{i=1}^n w_i y_i z_i=a\langle\mathbf{x},\mathbf{z}\rangle_w+b\langle\mathbf{y},\mathbf{z}\rangle_w\). By symmetry, linearity in the first argument also gives linearity in the second: \(\langle\mathbf{z},a\mathbf{x}+b\mathbf{y}\rangle_w=a\langle\mathbf{z},\mathbf{x}\rangle_w+b\langle\mathbf{z},\mathbf{y}\rangle_w\).

Finally, \(\langle\mathbf{x},\mathbf{x}\rangle_w=\sum_{i=1}^n w_i x_i^2\). Each term is nonnegative. If \(\mathbf{x}\ne\mathbf{0}\), at least one coordinate \(x_j\ne0\), so \(w_jx_j^2>0\), and hence the whole sum is strictly positive. If \(\mathbf{x}=\mathbf{0}\), every term is zero. Thus the product is positive definite and satisfies all the real inner product axioms.

For the reverse implication, suppose the formula defines an inner product. For each standard basis vector \(\mathbf{e}_i\), its \(i\)th coordinate is \(1\) and all other coordinates are \(0\), so \(\langle\mathbf{e}_i,\mathbf{e}_i\rangle_w=w_i\). Since \(\mathbf{e}_i\ne\mathbf{0}\), positive definiteness requires \(w_i>0\). This holds for every \(i\), proving the necessity as well as the sufficiency. \(\square\)

This argument checks the axioms directly; it does not depend on a special choice of dimension. The weights may differ, but they cannot vanish or be negative if the formula is to define an inner product.

Computing Weighted Products and Lengths

Weights can change lengths even when the vector itself has not changed. For example, \((1,2)^\top\) has standard squared length \(1^2+2^2=5\), but its squared length with weights \(2\) and \(5\) is \(22\). The difference comes from changing how strongly each coordinate contributes; it is not a change to the vector's coordinates.

Positive Weights as Coordinate Rescaling

There is another way to interpret the weighted product. Since each \(w_i>0\), its positive square root \(\sqrt{w_i}\) is a real number. Rescale coordinate \(i\) by this factor. The weighted product then becomes an ordinary dot product of the rescaled vectors.

Proof. Multiplication by \(D\) scales coordinate \(i\) of \(\mathbf{x}\) to \(\sqrt{w_i}x_i\), and coordinate \(i\) of \(\mathbf{y}\) to \(\sqrt{w_i}y_i\). By the definition of the standard dot product, \((D\mathbf{x})^\top(D\mathbf{y})=\sum_{i=1}^n(\sqrt{w_i}x_i)(\sqrt{w_i}y_i)=\sum_{i=1}^n w_i x_i y_i=\langle\mathbf{x},\mathbf{y}\rangle_w\). The middle equality uses \((\sqrt{w_i})^2=w_i\), which holds because \(w_i>0\). \(\square\)

The scaling viewpoint explains why positivity matters geometrically. A strictly positive scale changes the size of a coordinate without erasing or reversing it. If every coordinate is scaled this way, a nonzero vector remains nonzero. A zero weight would erase a coordinate entirely, while a negative weight would not have a real square root and would make the corresponding basis vector have negative squared length.

Weights Can Change Orthogonality

Two vectors are orthogonal with respect to a given inner product when their product is zero. Changing the weights can therefore change which pairs are orthogonal. In particular, weighted orthogonality need not match orthogonality under the standard dot product: each product uses a different rule for combining coordinate contributions.

In applications, weights can represent different units, priorities, or measurement scales across coordinates. A weighted product is useful only when those weights are chosen consistently with the intended geometry. For instance, calling two vectors orthogonal without specifying which product is being used can give an incomplete description.

Common Pitfall: Nonnegative Is Not Enough

A zero weight can destroy positive definiteness. It may seem sufficient to require that weights are nonnegative, since then every term \(w_i x_i^2\) is nonnegative. But an inner product requires a strictly positive self-product for every nonzero vector. With weights \(w_1=1\), \(w_2=0\), the nonzero vector \(\mathbf{x}=(0,1)^\top\) gives the incorrect conclusion that this formula defines an inner product merely because its terms are nonnegative. In fact,

This violates positive definiteness. The correction is to require \(w_i>0\) for every coordinate, not just \(w_i\ge0\). A negative weight fails as well: if \(w_j<0\), then \(\langle\mathbf{e}_j,\mathbf{e}_j\rangle_w=w_j<0\), which also violates positive definiteness.

When calculating, another practical safeguard is to keep each weight paired with its own coordinate product: the \(i\)th term must be \(w_i x_i y_i\). After computing, check symmetry by reversing the vectors, and check a claimed length by evaluating the self-product. These checks follow directly from the axioms and can expose a misplaced weight or sign error.