Throughout this work, V denotes a finite-dimensional real inner-product space. Unless explicitly stated otherwise, every vector entering the angular construction is normalized. These assumptions are essential. In a complex inner-product space, the inner product is generally complex and cannot be represented solely by a real angle, whereas for non-normalized vectors, the standard norm factors must be retained.

For an orthonormal basis { e i } i = 1 n ,

The off-diagonal zero summarizes two logically distinct statements: the indices differ, and the corresponding vectors are orthogonal. The purpose of the present notation is to keep these mechanisms separate.

Although the Kronecker delta and the Gram matrix already provide a complete mathematical description of orthogonality, they encode index identity and metric coupling in a compact form. The present decomposition does not replace either object. Instead, it separates their logical roles, allowing the geometric source of an off-diagonal contribution to remain explicit in calculations where orthogonality is created, tested, or lost.

The Kronecker delta encodes index identity and orthonormality in linear algebra and tensor notation [1][2]. For a general basis, inner products are collected in the Gram matrix [3][4]. To demonstrate utility beyond elementary examples, the manuscript develops two applications: a complete Gram-Schmidt calculation in ℝ 3 and a constrained-mechanics analysis of a homogeneous ladder sliding without friction. The latter distinguishes a purely kinematic divergence from the physically admissible motion governed by unilateral contact.

Definition 2.1(Off-diagonal selector). Define

The quantity Δ i j J is not an inner product. It only selects an off-diagonal index pair, indicating where an angular coupling may occur.

Let e i and e j be unit vectors, and let θ i j ∈ [ 0 , π ] denote their angle.

Definition 3.1 (Angular Kronecker-Jeno quantity).

Equivalently, δ i i KJ = 1 and δ i j KJ = cos θ i j for i ≠ j .

Proposition 3.2(Recovery of the classical Kronecker delta). If { e i } is orthonormal, then δ i j KJ = δ i j .

Proof. For i = j , the definition gives δ i i KJ = 1 . For i ≠ j , orthogonality gives θ i j = π / 2 , and therefore δ i j KJ = cos ( π / 2 ) = 0 . □

Theorem 4.1 (Structural nullity). Let i ≠ j and let e i , e j be unit vectors satisfying 〈 e i , e j 〉 = 0 .Then

Numerically, 0 J = 0 .

Proof. Because i ≠ j , one has 1 − δ i j = 1 . Orthogonality implies θ i j = π / 2 , that the angular factor vanishes. The symbol 0 J records the mechanism by which the ordinary numerical zero was obtained. □

Remark 4.2.The distinction is semantic and structural,not arithmetic.Standard algebra continues to use the ordinary real number zero.The label 0 J carries bookkeeping information only.

For

the inner product is

For normalized vectors, g i j = cos θ i j and g i i = 1 , so

Thus, symmetry, positive semidefiniteness, rank, and basis dependence are inherited directly from the Gram matrix.

Define the sign-independent angular-nullity matrix by

Unlike 1 − cos θ i j , this quantity is invariant under reversal of either vector and reaches its maximum precisely at orthogonality.

For arbitrary nonzero vectors,

Consequently, the angular definition does not reproduce the full Gram matrix unless ‖ e i ‖ = 1 for every i . Restoring the norm factors gives the standard inner product. In particular, for i ≠ j ,

The proposal, therefore, does not replace the ordinary inner product. In complex spaces, a real angle alone also loses phase information; a direct extension would require a phase-sensitive construction and lies outside the present scope.

The Gram-Schmidt process and its numerically stable variants are standard tools in matrix computation and numerical linear algebra [5][6]. Let B = ( v 1 , ⋯ , v n ) be linearly independent. Gram-Schmidt constructs

where

Thus, the coefficient removed at each step is exactly the off-diagonal angular coupling with a previously normalized direction.

For a normalized ordered family B = ( b 1 , ⋯ , b n ) , define

Proposition 7.1.For normalized vectors,

and N ( B ) = ( n 2 ) if and only if the family is pairwise orthogonal.

Proof. Every term belongs to [ 0 , 1 ] . The upper bound is reached exactly when cos θ i j = 0 for every pair i < j . □

Remark 7.2 (Interpretation). Gram-Schmidt may be interpreted as a sequential elimination of the off-diagonal Gram couplings associated with the already constructed orthonormal directions.The final orthonormal family hasaGram matrix I n and maximizes N .The functional is not claimed to be monotone for every possible intermediate normalization convention; rather,it provides a global diagnostic of pairwise orthogonality for each normalized family being compared.

The frictionless sliding-ladder problem is a classic example of constrained rigid-body mechanics. The derivation below follows the conventional Newtonian and energy formulations described in standard mechanics texts [7]-[9], and then interprets the relevant geometric factors through the angular decomposition.

Consider a homogeneous rigid ladder of length L and mass m , with its lower endpoint on a frictionless horizontal floor and its upper endpoint on a frictionless vertical wall. Let θ ( t ) be the angle measured from the floor. While both contacts are maintained,

Table 1 summarizes the interpretive role of the proposed decomposition. The right-hand column does not add new dynamics or algebra; it makes explicit which geometric factor is active in each standard calculation.

Table 1. Classical objects and their angular interpretation.

The Gram-Schmidt application shows that the notation can track which off-diagonal coupling is removed at each projection step. This is mathematically equivalent to standard orthogonalization, but it preserves the angular meaning of the projection coefficient. The functional N ( B ) provides a sign-independent scalar diagnostic of pairwise orthogonality for normalized families.

The ladder application goes further by separating two issues that are often conflated. First, the kinematic relation y ˙ = − ( cos θ / sin θ ) x ˙ contains a denominator that vanishes when the ladder approaches the horizontal configuration. Second, the freely released ladder does not remain in two-point contact until that configuration. The unilateral contact condition forces the two-contact model to terminate earlier, at sin θ s = ( 2 / 3 ) sin θ 0 . The angular decomposition identifies the geometric factor that vanishes, whereas the contact inequality determines the physically admissible interval.

The condition N w ≥ 0 is not encoded by δ i j KJ itself. It remains a mechanical contact condition. Accordingly, the usefulness of the angular decomposition is diagnostic and interpretive rather than dynamical. The present results do not establish a new tensor, scalar product, or force law. They show that a standard Gram entry can be decomposed into index selection and angular coupling, and that this bookkeeping can clarify both orthogonalization and constrained mechanics.

The framework remains restricted to normalized real vectors unless the usual norm factors are restored. In complex inner-product spaces, phase information cannot be represented by a single real angle. These limitations are not technical afterthoughts; they define the domain in which the proposed notation is mathematically equivalent to familiar objects.

The angular quantity

recovers the classical Kronecker delta in an orthonormal basis and coincides with the Gram matrix in a normalized real basis. Its role is interpretive: it separates index identity from the angular mechanism responsible for an off-diagonal metric contribution.

Two concrete uses were demonstrated. In Gram-Schmidt orthogonalization, off-diagonal Gram couplings are sequentially removed until the Gram matrix becomes the identity, while N ( B ) reaches its maximum value. In the sliding-ladder problem, the apparent infinite endpoint speed follows from division by the vanishing vertical coupling sin θ under an incompatible constant-speed constraint. In the physically released system, the ladder loses wall contact earlier, at sin θ s = ( 2 / 3 ) sin θ 0 ; for release from the vertical, y s = 2 L / 3 .

Thus, the decomposition does not replace classical linear algebra or mechanics. It makes explicit which angular factor vanishes and where a constrained model ceases to be physically admissible. Future work may examine whether the same bookkeeping is useful in constrained dynamics, numerical orthogonalization, finite-element formulations, and tensor calculations in which orthogonality evolves during computation.

Let B = ( e 1 , ⋯ , e n ) be a normalized family in a finite-dimensional real inner-product space. Its Gram matrix is G = ( g i j ) , where

For i = j , normalization gives g i i = 1 . For i ≠ j , the real inner-product identity gives g i j = cos θ i j . Hence

Therefore, δ KJ and G are identical matrices under the stated assumptions. Every algebraic property used in this manuscript is consequently inherited from the Gram matrix rather than postulated independently.