4.14.4 Tensor Scalar Pullout Rule
The Tensor Scalar Pullout Rule allows scalars to commute with tensors, simplifying operations in multilinear algebra and tensor calculus.
Tensor Scalar Pullout Rule is the practical computational rule, derived directly from the tensor multilinear homogeneity property, stating that a scalar factor multiplying any single argument of a tensor can be moved outside the entire evaluation, multiplying the whole result instead. It is the everyday working form of homogeneity, used constantly when simplifying tensor expressions, and it applies uniformly no matter which slot the scalar originally appeared in or how many other scalars are already pulled out.
Statement of the Rule
Pulling a Scalar from a Single Slot
For a type (p, q) tensor T on a vector space V, if any one argument in the input tuple is written as a scalar λ times a vector or covector, that scalar can be pulled entirely outside the evaluation:
This is precisely a restatement of homogeneity in that slot, phrased as a directive: whenever a scalar multiplies one of the arguments, rewrite the expression by moving the scalar out front and evaluating the tensor on the un-scaled argument instead.
Pulling Scalars from Several Slots at Once
If several arguments each carry their own scalar factor, the rule can be applied slot by slot, once for each scaled argument, and the pulled-out scalars combine by ordinary multiplication:
allowing every scalar factor present in an input tuple to be gathered together in front of a single evaluation on the un-scaled arguments.
Applying the Rule in Component Computations
Simplifying Before Expanding in a Basis
When an argument is presented as a scalar multiple of a basis vector, such as x = c e_i, the scalar pullout rule allows the scalar c to be pulled out before the remaining evaluation is carried out on the single basis vector e_i, which is often simpler than expanding the argument fully and applying the component evaluation formula directly to a general coordinate array.
Interaction with the Basis Expansion Formula
The scalar pullout rule is one ingredient used repeatedly, together with additivity, in deriving the tensor vector valued basis expansion and the ordinary component evaluation formula: each term in those expansions arises from pulling out the scalar coordinate of a basis expansion before evaluating T on the corresponding basis vectors, and it is this repeated pullout that produces the coefficient in front of each basis evaluation term.
Common Uses of the Rule
Normalizing Arguments Before Evaluation
The rule is frequently used to normalize an argument, dividing out its own scale by writing x = |x| · (x / |x|) and pulling |x| outside the evaluation, so that the remaining evaluation involves only a unit vector, which can simplify subsequent analysis of how a tensor depends on the direction of an argument independently of its magnitude.
Simplifying Expressions Involving Repeated Scaling
When a tensor expression is built up through several intermediate steps, each possibly introducing its own scalar factor, such as differentiating or rescaling physical quantities, the scalar pullout rule allows all such factors to be collected together at the end of the computation rather than tracked separately through each intermediate step, since homogeneity guarantees they can always be factored out consistently.
Limits of the Rule
The Rule Applies to a Single Argument at a Time
The scalar pullout rule pulls a scalar out of one argument in one slot; it does not permit pulling out a common factor that is shared, in a more complicated way, across the internal structure of a single argument that is not itself a simple scalar multiple, since homogeneity is stated with respect to scalar multiplication of an entire argument, not with respect to arbitrary internal rescalings of its components.
Does Not Apply to Nonlinear Combinations
If an expression involves an argument raised to a power, or multiplied against itself, the scalar pullout rule does not apply directly in the simple linear form given above, since such an expression would not correspond to a linear dependence on that argument in the first place, and the tensor multilinear homogeneity property is stated only for arguments appearing to the first power in each slot.
Diagrammatic Summary
The diagram illustrates the scalar pullout rule as the migration of a scalar factor from multiplying a single argument inside the tensor to multiplying the entire evaluated result, licensed directly by the homogeneity property of that slot.