12.5.2 Tensor Scalar Multiplication Component Scaling

Tensor Scalar Multiplication Component Scaling is the mechanism by which every individual scalar component of a tensor is multiplied by one common scalar factor, applying that same factor uniformly across all index positions without altering the relative proportions among the components.

Mechanics of Component Scaling

Uniform Application Across Indices

For a tensor $$A$$ of type $$(p,q)$$ and a scalar $$k$$, component scaling produces the tensor $$kA$$ whose entry at any fixed index choice is:

Every one of the components, no matter how many indices the tensor carries or what values those indices take, is multiplied by the identical scalar $$k$$.

Contrast with Non-Uniform Scaling

Component scaling specifically excludes any scheme in which different components would be multiplied by different factors. Multiplying, for instance, the components with a particular fixed first index by one scalar and the remaining components by another scalar would not constitute tensor scalar multiplication and would generally destroy the tensor's transformation properties.

Effect on Simple Examples

Scaling a Vector

For a vector $$v$$ with components $$v^i$$, component scaling by $$k$$ produces a new vector with components:

Each entry of the vector is stretched or shrunk by the same factor, preserving the vector's direction when $$k$$ is positive and reversing it when $$k$$ is negative.

Scaling a Rank-2 Tensor

For a rank-2 tensor with components $$A_{ij}$$, component scaling produces:

with every entry of the two-dimensional array of components scaled by the same amount.

Preserved and Altered Quantities

What Component Scaling Preserves

Component scaling preserves the ratios between different components of the tensor, since every component is multiplied by the same factor. It also preserves the tensor's rank, variance, and dimensionality, since scaling does not touch the index structure at all.

What Component Scaling Alters

Component scaling alters the overall magnitude of the tensor's components. When applied with a negative scalar, it also reverses the sign of every component. When applied with a scalar whose absolute value is less than one, it shrinks the components; when greater than one, it enlarges them.

Consistency Under Change of Basis

Scaling Commutes with Transformation

If the components of $$A$$ are transformed to a new basis and then scaled by $$k$$, the result matches scaling first and transforming afterward, because the transformation law is linear in the components. This ensures that component scaling represents a genuine operation on the underlying abstract tensor, consistently observed no matter which basis is used to compute the components.

Illustration