A New Proof of the Pythagorean Theorem and Its Application to Element Decompositions in Topological Algebras

Abstract

We present a new proof of the Pythagorean theorem which suggests a particular decomposition of the elements of a topological algebra in terms of an “inverse norm” (addressing unital algebraic structure rather than simply vector space structure). One consequence is the unification of Euclidean norm, Minkowski norm, geometric mean, and determinant, as expressions of this entity in the context of different algebras.

1. Introduction

Apart from being unital topological *-algebras, matrix algebras, special Jordan algebras and Cayley-Dickson algebras would seem to have little else in common. For example, the matrix algebra ℝ^𝑛×𝑛 is associative but noncommutative, the special Jordan algebra derived from it is nonassociative but commutative (and so are the spin factor Jordan algebras), and Cayley-Dickson algebras are both nonassociative and noncommutative (apart from the three lowest dimensional instances). However, these latter sets of algebras share an interesting feature. Each is associated with a function 𝑓(𝑠) that vanishes on the nonunits and provides a decomposition of every unit as 𝑠=𝑓(𝑠)∇^∗𝑓𝑠⁻¹, (1.1) with 𝑓∇^∗𝑓𝑠⁻¹=1, (1.2) where ∇^∗𝑓(𝑠⁻¹) indicates that the gradient ∇𝑓 is evaluated at 𝑠⁻¹ following which the involution * is applied. For example, (i)on an 𝑛-dimensional Cayley-Dickson algebra, 𝑓(𝑠) is the quadratic mean of the components multiplied by √𝑛, that is, 𝑓(𝑠) is the Euclidean norm, (ii)on the algebra ℝ^𝑛 with component-wise addition and multiplication, 𝑓(𝑠) is the geometric mean of the absolute values of the components multiplied by √𝑛, (iii)on the matrix algebra ℝ^𝑛×𝑛, 𝑓(𝑠) is the (principal) 𝑛th root of the determinant multiplied by √𝑛, (iv)on the spin factor Jordan algebras, 𝑓(𝑠) is the Minkowski norm. Looked at another way, the Euclidean norm, the geometric mean, the 𝑛th root of the determinant, and the Minkowski norm are all expressions of the same thing in the context of different algebras. With respect to topological *-algebras, this “thing” supercedes the Euclidean norm and determinant since neither is meaningful in all settings for which the solution to (1.1) and (1.2) is meaningful.

There is another aspect relevant to the Cayley-Dickson algebras. In addition to (1.1) and (1.2), there is a function 𝑓 on the elements of the algebra such that for any unit 𝑠 in the algebra, 𝑠=𝑓(𝑠)∇𝑓(𝑠), (1.3) with 𝑓∇𝑓(𝑠)=1. (1.4) This equation set makes no reference to multiplicative structure; that is, it is a general property of the underlying vector space. Indeed, 𝑓(𝑠) is again the Euclidean norm. In fact, (1.3) and (1.4) can be derived from the (Hilbert formulation) axioms of plane Euclidean geometry without use of the Pythagorean theorem—and as such can be used as the centerpiece of a new proof of that theorem.

So, we first prove the Pythagorean Theorem by deriving (1.3) and (1.4), we then use the latter equations to develop (1.1) and (1.2), following which we demonstrate the assertions of the first paragraph of this Introduction. Ultimately, existence of the decomposition (1.1), (1.2) is forwarded as a kind of surrogate for the Pythagorean theorem in the context of topological algebras.

It will also be seen that there is a hierarchy related to the basic equations, evidenced by progressively more structure accompanying the solution function on particular algebras. The equations (1.1) and (1.2) and have a clear analogy with the form of (1.3) and (1.4). Both cases present a decomposition of the units of an algebra as the product of a particular function's value at that point multiplied by a unity-scaled orientation point dependent on the function's gradient. The equations are nonlinear, in general. However, along with the prescription that 𝑓(𝑠) vanishes on the nonunits, the above particular matrix, Cayley-Dickson, and spin factor Jordan algebras, also happen to satisfy the additional property that there is a real constant 𝛼 such that for any unit 𝑠 in the algebra, 𝑓(𝑠)𝑓𝑠⁻¹=𝛼. (1.5) Replacing 𝑠 with 𝑠⁻¹ in (1.1), the above implies a decomposition of the inverse of a point as 𝑠⁻¹∇=𝛼^∗𝑓(𝑠)𝑓(𝑠). (1.6) Furthermore, multiplying both sides of (1.6) by 𝑓(𝑠)𝑠 indicates that the above is a linear equation for 𝑓. One can be even more restrictive and consider the set of algebras on which a function exists satisfying all of the above where in addition (1.5) is strengthened to 𝑓(𝑠₁)𝑓(𝑠₂)=𝛼𝑓(𝑠₁𝑠₂) for any two units 𝑠₁,𝑠₂. In this case, the units form a group and 𝑓(𝑠)/𝛼 is a homomorphism on this group. The octonions occupy a special place as an algebra satisfying this prescription that is not a matrix subalgebra.

2. Euclidean Decomposition

Theorem 2.1 (Pythagoras). In a space satisfying the axioms of plane Euclidean geometry, the square of the hypotenuse of a right triangle is equal to the sum of the squares of its two other sides.

The theorem hypothesis is assumed to indicate the Hilbert formulation of plane Euclidean geometry [1]. One will refer to the point set in question as the “Euclidean plane.” Points will be denoted by lower case Roman letters and real numbers by lower case Greek letters.

Proof. We begin by providing an outline of the proof.

A vector space structure is defined on the Euclidean plane 𝐸 after identifying one of the vertices of the hypotenuse of the given right triangle as an origin 𝑜. We define the Euclidean norm implicitly as the function 𝑓∶𝐸→ℝ giving the length of a line segment from the origin to any given point in the plane. Since the Hilbert formulation includes continuity axioms, we can employ the usual notions relating to limits and thereby define directional derivatives. The crux of the proof, Lemma 2.3, is the demonstration that the parallel axiom implies the existence and continuity of the directional derivatives at points other than the origin, and the largest directional derivative at a point 𝑠 associated with a unit length direction line segment has unit value and is such that its direction line segment is collinear with line segment 𝑜𝑠. The novelty lies in the necessity that this be accomplished in the absence of an explicit formula for the Euclidean norm. A Cartesian axis system is now generated from the two sides of the given triangle forming the right angle, following which an isomorphism from 𝐸 to vector space ℝ² is easily demonstrated. Using this isomorphism, Lemma 2.3 is seen to imply the existence of the gradient ∇𝑓(𝑡)∈𝐸 for 𝑡 not at the origin, with its characteristic property regarding generation of a directional derivative from a particular specified direction line segment, and such that the origin, 𝑡, and ∇𝑓(𝑡) are collinear. It is then a simple matter to show 𝑡=𝑓(𝑡)∇𝑓(𝑡) and 𝑓(∇𝑓(𝑡))=1. For 𝑡=(𝜏₁,𝜏₂), a solution to the latter partial differential equation is supplied by 𝑓(𝜏₁,𝜏₂)=𝜏²₁+𝜏²₂. This solution is unique because 𝑡=𝑓(𝑡)∇𝑓(𝑡) implies that 𝑡 and ∇𝑓(𝑡) are collinear with the origin, so that the equation can be written as an ordinary differential equation—which is easily shown to have a unique solution. This explicit representation of the Euclidean norm implies the Pythagorean theorem, thus concluding the proof.

We now fill in details the above argument.

Given a right triangle Δ{𝑜,𝑎,𝑠} with line segment 𝑜𝑠 as the hypotenuse, we define a function 𝑓(𝑡) that gives the length of the line segment 𝑜𝑡 for any point 𝑡 in the plane, with the convention that 𝑓(𝑜)=0. The continuity axioms that are part of the Hilbert formulation support the equivalent of the least upper bound axiom for the real number system, and we have the usual continuity properties related to ℝ, whose elements identify lengths of line segments in particular. Thus, the usual notion of limit can be defined and is assumed.

Definition 2.2. A direction line segment with respect to a particular point is any line segment one of whose endpoints is the given point. If the following limit exists, the directional derivative of 𝑓 at 𝑡 with respect to direction line segment 𝑡𝑤 is 𝐷_𝑡𝑤𝑓(𝑡)≡lim_𝜖→0𝑓𝑡⊕𝜖𝑡𝑤−𝑓(𝑡)𝜖, (2.1) where 𝑡⊕𝜖𝑡𝑤 formally denotes a point on the line containing 𝑡𝑤 such that the line segment defined by this point and 𝑡 has length given by the product of |𝜖| and the length of 𝑡𝑤, and 𝑡 lies between this point and 𝑤 if and only if 𝜖<0.

Lemma 2.3. For any point w different from s, the directional derivative of f at s specified by sw exists and is continuous at s. Furthermore, the largest directional derivative at s associated with a direction line segment sw of a particular fixed length is such that o,s,w are collinear with s between o and w, and if the direction line segment has unit length, then the largest directional derivative has unit value.

Proof of Lemma 2.3. Let the length of 𝑠𝑤 be 𝜆. If 𝑤 lies on the line containing 𝑜𝑠 with 𝑠 between 𝑜 and 𝑤, it is clear that the directional derivative exists and has a value given by 𝜆, since the expression inside the limit in (2.1) has this value for all 𝜖≠0. In the same way, we can demonstrate that the directional derivative exists when 𝑤 is on the line containing 𝑜𝑠 but 𝑠 is not between 𝑜 and 𝑤—in which case the directional derivative has value −𝜆.
Thus, suppose 𝑠𝑤 is not on the line containing 𝑜𝑠. Figure 1 can be used to keep the following constructions in context. Consider the orthogonal projection of the point 𝑠⊕𝜖𝑠𝑤 to the line containing 𝑜𝑠, that is, the point 𝑝 on the line containing 𝑜𝑠 such that ∠{𝑜,𝑝,𝑠⊕𝜖𝑠𝑤} is a right angle (the expression ∠{𝑜,𝑝,𝑠⊕𝜖𝑠𝑤} denotes the angle within the triangle Δ{𝑜,𝑝,𝑠⊕𝜖𝑠𝑤} that is formed by the two line segments 𝑜𝑝 and 𝑝(𝑠⊕𝜖𝑠𝑤)). Consider the circle centered at 𝑜 of radius 𝑓(𝑠⊕𝜖𝑠𝑤). We claim that 𝑝 is in the region enclosed by the circle, that is, 𝑓(𝑝)<𝑓(𝑠⊕𝜖𝑠𝑤). Indeed, suppose this is false. Consider the circle with center at 𝑜 of radius 𝑓(𝑝), and let the tangent line to this circle at the point 𝑝 be denoted 𝑇_𝑝. Being a tangent, all of the points 𝑦 of 𝑇_𝑝 other than 𝑝 will be such that 𝑓𝑦>𝑓𝑝≥𝑓(𝑠⊕𝜖𝑠𝑤). (2.2) Note that 𝑇_𝑝 intersects the line containing 𝑜𝑠 at a right angle (a tangent to a circle at a particular point is perpendicular to the circle radius at that point). But there is only one line through 𝑝 that meets the line containing 𝑜𝑠 at a right angle, and that is the line containing 𝑝(𝑠⊕𝜖𝑠𝑤), as previously defined. So 𝑇_𝑝 must contain 𝑝(𝑠⊕𝜖𝑠𝑤), that is, 𝑠⊕𝜖𝑠𝑤 is on 𝑇_𝑝 but is not the point 𝑝. The first inequality of (2.2) would then imply that 𝑦=𝑠⊕ϵ𝑠𝑤∈𝑇_𝑝 is such that 𝑓(𝑠⊕𝜖𝑠𝑤)>𝑓(𝑝). This contradicts the second inequality of (2.2). Hence, we have established our claim that 𝑓(𝑝)<𝑓(𝑠⊕𝜖𝑠𝑤).
Let 𝑃_𝑜𝑠 be the line perpendicular to 𝑜𝑠 at 𝑜. Since 𝑠 is not the point 𝑜, the line containing 𝑠𝑤 intersects 𝑃_𝑜𝑠 in at most one point. In what follows, we assume |𝜖| is small enough so that no point of 𝑠(𝑠⊕𝜖𝑠𝑤) is on 𝑃_𝑜𝑠. Again consider the circle centered at 𝑜 with diameter 𝑓(𝑠⊕𝜖𝑠𝑤). Its tangent at 𝑠⊕𝜖𝑠𝑤 must intersect the line containing 𝑜𝑠 at the point 𝑟 (since 𝑠⊕𝜖𝑠𝑤 is not on 𝑃_𝑜𝑠, the tangent cannot be parallel to 𝑜𝑠). Being on a tangent but not the point of tangency itself, 𝑟 is necessarily external to the region enclosed by the circle (𝑓(𝑟) is greater than the circle radius). This circle intersects the line containing 𝑜𝑠 at two points defining a diameter of the circle. We denote as 𝑞 the one of these two points such that 𝑞 is between 𝑝 and 𝑟.
We are first required to show the existence of the limit in (2.1) (for 𝑡=𝑠), which we can write as lim_𝜖→0(𝑓(𝑞)−𝑓(𝑠))/𝜖, since 𝑞 and 𝑠⊕𝜖𝑠𝑤 both lie on the aforementioned circle. To do this, we will initially assume that 𝜖>0 and show that lim_𝜖→0+(𝑓(𝑞)−𝑓(𝑠))/𝜖 exists, after which it will be clear that an entirely analogous argument establishes the same value for lim_𝜖→0−(𝑓(𝑞)−𝑓(𝑠))/𝜖.
We have 𝑓𝑞−𝑓(𝑠)=𝑓𝑞𝑝𝜖𝜆−𝑓+𝑓𝑝(𝑠)𝜖𝜆−𝑓,𝑓𝑞𝑝𝜖𝜆−𝑓≤𝑓(𝑟)𝑝𝜖𝜆−𝑓,𝜖𝜆 (2.3)(2.4) since 𝑞 is between 𝑝 and 𝑟. Note that 𝜖𝜆 is the length of 𝑠(𝑠⊕𝜖𝑠𝑤), according to Definition 2.2. The triangle Δ{𝑝,𝑜,𝑠⊕𝜖𝑠𝑤} is similar to Δ{𝑝,𝑠⊕𝜖𝑠𝑤,𝑟} (ultimately, because the tangent line to the circle at 𝑠⊕𝜖𝑠𝑤 implies that ∠{𝑜,𝑠⊕𝜖𝑠𝑤,𝑟} is a right angle). Let 𝜇 be the length of 𝑝(𝑠⊕𝜖𝑠𝑤). It follows that 𝑓(𝑟)−𝑓𝑝𝜇=𝜇𝑓𝑝. (2.5)
If 𝑠 and 𝑝 are ever the same point for some value of 𝜖, then 𝑠(𝑠⊕𝜖𝑠𝑤) is perpendicular to 𝑜𝑠, and will remain so for any other 𝜖, so that 𝑠 and 𝑝 will always be the same point. In that case, 𝑓(𝑠)=𝑓(𝑝), and 𝜇=𝜖𝜆. It then follows that the right-hand side of (2.5) tends to zero with 𝜖 (since this right-hand side is (𝜖𝜆)/𝑓(𝑠)), so the right-hand side of (2.4) also tends to zero with 𝜖, which means that the first term on the right-hand side of (2.3) tends to zero with 𝜖. But since 𝑓(𝑠)=𝑓(𝑝), the second term on the right-hand side of (2.3) is zero. It then follows that lim_𝜖→0(𝑓(𝑞)−𝑓(𝑠))/𝜖=0 and, in particular, the required limit exists.
Thus, suppose 𝑠 and 𝑝 are different, and consider Δ{𝑝,𝑠,𝑠⊕𝜖𝑠𝑤}. This defines a set of similar triangles for all values of 𝜖≠0, because the angle ∠{𝑝,𝑠,𝑠⊕𝜖𝑠𝑤} does not change as 𝜖 varies and ∠{𝑠,𝑝,𝑠⊕𝜖𝑠𝑤} remains a right angle. Consequently, 𝜇/(𝜖𝜆) is a nonzero constant for all 𝜖>0 since this is the ratio of two particular sides of each triangle in this set of similar triangles. This means that lim_𝜖→0+𝜇=0, since 𝜇/(𝜖𝜆) could not otherwise remain a constant because 𝜖𝜆 tends to zero with 𝜖. On the other hand, because |𝑓(𝑝)−𝑓(𝑠)|/(𝜖𝜆) is also constant as 𝜖 varies (being a ratio of a different combination of sides of these same triangles), it must also follow that lim_𝜖→0+𝑓(𝑝)=𝑓(𝑠)≠0 (which also means that 𝑜 is not between 𝑝 and 𝑠 for small 𝜖). So, it must be that the right-hand side of (2.5) tends to zero as 𝜖>0 tends to zero. Consequently, the right-hand side of (2.4) also tends to zero as 𝜖>0 tends to zero (because, as we have noted, 𝜇/(𝜖𝜆) is constant as 𝜖>0 varies). Hence, the first term on the right-hand side of (2.3) also tends to zero as 𝜖>0 becomes small. This means that the left-hand side of (2.3) has the same limit as 𝜖>0 tends to zero as the second term on the right-hand side of (2.3) (assuming the limit exists). But, we have already noted that the term |𝑓(𝑝)−𝑓(𝑠)|/(𝜖𝜆) is a constant as ϵ varies, since it is determined by the ratio of particular sides of Δ{𝑝,𝑠,𝑠⊕𝜖𝑠𝑤}, and (as we have noted) for any 𝜖≠0 all such triangles are similar. In fact (removing the absolute value sign in the numerator), we further claim that 𝑓𝑝(𝑠)𝜅≡−𝑓𝜖𝜆 (2.6) is a constant for all small 𝜖>0. To see this, recall that for small 𝜖>0, 𝑜 is not between 𝑠 and 𝑝, and ∠{𝑝,𝑠,𝑠⊕𝜖𝑠𝑤} is constant. Now, if 𝑝 changes from being between 𝑜 and 𝑠 versus not being between 𝑜 and 𝑠 as 𝜖>0 varies, the latter angle must change from being an acute angle to being an obtuse angle, or vice versa, which contradicts the fact that the angle is constant for all 𝜖>0. For small 𝜖>0, it follows that 𝑠 is always between 𝑜 and 𝑝, or 𝑝 is always between 𝑜 and 𝑠—which establishes our claim that (2.6) is constant. Therefore, lim_𝜖→0+𝑓𝑞−𝑓(𝑠)𝜖=𝜆𝜅. (2.7) An analogous argument establishes lim_𝜖→0−(𝑓(𝑞)−𝑓(𝑠))/𝜖=𝜆𝜅, because Δ{𝑝,𝑠,(𝑠⊕𝜖𝑠𝑤)} are similar triangles for all nonzero values of 𝜖. This establishes the limit in (2.1).
Next, we establish the continuity of each directional derivative. Consider any sequence of line segments {𝑠_𝑖𝑤_𝑖} such that {𝑠,𝑤,𝑤_𝑖,𝑠_𝑖} form a parallelogram for all 𝑖, and the limit of the length of the line segments {𝑠_𝑖𝑠} is zero (which implies that the limit of the length of the line segments {𝑤_𝑖𝑤} is also zero). To establish the continuity of a directional derivative at 𝑠 it is required to show that the left-hand-side of the following equation is zero: lim_𝑖→∞𝐷_𝑠𝑤𝑓(𝑠)−𝐷_{𝑠𝑖𝑤𝑖}𝑓(𝑠)=lim_𝑖→∞𝜅−𝜅_𝑖𝜆. (2.8) For each 𝑖, 𝜅_𝑖 is the ratio of particular sides of any member of a particular set of similar triangles, as is also the case for 𝜅 as above. Furthermore, as 𝑖 increases, the ratio of the lengths of the sides of the triangle relevant to 𝜅_𝑖 converges to the same ratio as that of the corresponding triangles relevant to 𝜅, since 𝑠_𝑖 converges to 𝑠 and 𝑤_𝑖 converges to 𝑤. Therefore, |𝐷_𝑠𝑤𝑓(𝑠)−𝐷_{𝑠𝑖𝑤𝑖}𝑓(𝑠)| necessarily tends to zero, implying continuity of the directional derivative at 𝑠.
We next show that the largest directional derivative at 𝑠 associated with a direction line segment 𝑠𝑤 of a particular fixed length is such that 𝑜,𝑠,𝑤 are collinear with 𝑠 between 𝑜 and 𝑤. We have from (2.7) (and the equation in the following sentence) that the directional derivative is 𝜆𝜅. Thus, we only need to show that |𝑓(𝑝)−𝑓(𝑠)|≤𝜖𝜆, with equality if and only if 𝑠𝑤 is on the line containing 𝑜𝑠, because once that is established it is easy to show that if 𝑜,𝑠,𝑤 are collinear, then 𝜅 will be negative if 𝑠 is not between 𝑜 and 𝑤, and will be positive otherwise.
Now, 𝜆𝜖 is the length of the hypotenuse of right triangle Δ{𝑠,𝑝,𝑠⊕𝜖𝑠𝑤}, and |𝑓(𝑝)−𝑓(𝑠)| is the length of the side that is on the line containing 𝑜𝑠. So, we only need to show that a nonhypotenuse side of a right triangle has a shorter length than the hypotenuse. Thus, consider any right triangle Δ{𝑎,𝑏,𝑐} with ∠{𝑎,𝑏,𝑐} being the right angle (vertex 𝑎 here has nothing to do with vertex 𝑎 of our earlier given right triangle Δ𝑜,𝑎,𝑠). Consider a circle centered at 𝑎 having radius 𝑎𝑐. This circle intersects the line containing 𝑎𝑏 at a point 𝑏^′. Suppose the length of 𝑎𝑏^′ is less than the length of 𝑎𝑏, meaning that the length of the hypotenuse is smaller than the length of one of the other sides. Then 𝑏 is external to the circle. So, consider a second circle centered at 𝑎 but now with radius given by the length of 𝑎𝑏. This circle intersects the line containing 𝑎𝑏 at the point 𝑏. The two circles are concentric, with the circle containing 𝑏 lying wholly external to the circle containing 𝑐. Now, the second circle (containing 𝑏) has a tangent at 𝑏 making a right angle with the line containing 𝑎𝑏 (tangents are perpendicular to the radius at the point of tangency). But the line containing 𝑏𝑐 is also perpendicular to the line containing 𝑎𝑏 (since ∠{𝑎,𝑏,𝑐} is already given as a right angle). So the point 𝑐 (which lies on our first circle of radius given by the length of 𝑎𝑐) must also be a point on the tangent line at the point 𝑏 of our second circle. This is impossible since, given two concentric circles, a tangent to a point on the circle of greater radius cannot intersect the circle of smaller radius. This contradicts the assumption that the hypotenuse is smaller than the length of one of the sides. Furthermore, the hypotenuse cannot equal the length of one of the other sides of the right triangle, because in that case a circle centered at 𝑎 with radius equal to the length of the hypotenuse would intersect the right triangle at the two points 𝑏 and 𝑐, which would require that a tangent line to the circle at 𝑏 (making a right angle with the line containing 𝑎𝑏) would have another point of intersection with the circle (i.e., at 𝑐, again since the line containing 𝑏𝑐 is also perpendicular to the line containing 𝑎𝑏 and there can be only one such perpendicular). Thus, unless 𝑠,𝑝,𝑠⊕𝜖𝑠𝑤 are collinear, |𝑓(𝑝)−𝑓(𝑠)|<𝜖𝜆, and it is immediately verified that |𝑓(𝑝)−𝑓(𝑠)|=𝜖𝜆 if the points are collinear. It is then evident that the value of the largest directional derivative is 𝜆. Thus, if 𝜆=1, the largest directional derivative has unit value. Hence, Lemma 2.3 is proved.

Figure 1 

Diagram relating to proof that the function giving the length of a line segment with one endpoint at 𝑜 has continuous directional derivatives at points other than 𝑜, Lemma 2.3. Three facts used in the proof are as follows: (1) Δ{𝑝,𝑜,𝑠⊕𝜖𝑠𝑤} is similar to Δ{𝑝,𝑠⊕𝜖𝑠𝑤,𝑟} because ∠{𝑜,𝑠⊕𝜖𝑠𝑤,𝑟} is a right angle, (2) for all 𝜖≠0, the triangles Δ{𝑝,𝑠,𝑠⊕𝜖𝑠𝑤} are similar, (3) 𝑞 is between 𝑝 and 𝑟.

Definition 2.2 actually suggests two operations, and these will be referred to in Lemma 2.5 as “the Definition 2.2 associated operations.” That is, we define the (more general) expression 𝑐⊕𝛾𝑢𝑣 to represent 𝑐⊕𝛾𝑣⊕(−𝛾)𝑢 subject to the following. (I)The multiplication of a scalar with a point, 𝛾𝑣, is defined to be the point on the line containing 𝑜𝑣 such that 𝑓(𝛾𝑣)=|𝛾|𝑓(𝑣) and such that 𝑜 is between this new point and 𝑣 if and only if 𝛾 is negative. (II)The “sum” of two points, ℎ₁⊕ℎ₂, is defined as follows. (i)If 𝑜,ℎ₁,ℎ₂ are not collinear, ℎ₁⊕ℎ₂ is defined to be the point 𝑧 such that the vertices {𝑜,ℎ₁,ℎ₂,𝑧} form a parallelogram.(ii)If either ℎ₁ or ℎ₂ is the point 𝑜, then ℎ₁⊕ℎ₂ is the point that is not 𝑜, or is 𝑜 if ℎ₁ and ℎ₂ are both 𝑜.(iii)If ℎ₁ and ℎ₂ are the same point, then ℎ₁⊕ℎ₂ is the point on the line containing 𝑜ℎ₁ such that 𝑓(ℎ₁⊕ℎ₂)=2𝑓(ℎ₁) and 𝑜 is not between ℎ₁ and ℎ₁⊕ℎ₂.(iv)If 𝑜,ℎ₁,ℎ₂ are distinct and collinear, (1)when 𝑜 is not between ℎ₁,ℎ₂ then ℎ₁⊕ℎ₂ is the point on the line containing ℎ₁ℎ₂ such that 𝑓(ℎ₁⊕ℎ₂)=𝑓(ℎ₁)+𝑓(ℎ₂) and 𝑜 is not between ℎ₁ and ℎ₁⊕ℎ₂, (2)when 𝑜 is between ℎ₁,ℎ₂, (a)if 𝑓(ℎ₂)>𝑓(ℎ₁) then ℎ₁⊕ℎ₂ is the point on the line containing ℎ₁ℎ₂ such that 𝑓(ℎ₁⊕ℎ₂)=𝑓(ℎ₂)−𝑓(ℎ₁) and 𝑜 is between ℎ₁ and ℎ₁⊕ℎ₂, (b)if 𝑓(ℎ₂)<𝑓(ℎ₁) then ℎ₁⊕ℎ₂ is the point on the line containing ℎ₁ℎ₂ such that 𝑓(ℎ₁⊕ℎ₂)=𝑓(ℎ₁)−𝑓(ℎ₂) and 𝑜 is between ℎ₂ and ℎ₁⊕ℎ₂, (c)if 𝑓(ℎ₁)=𝑓(ℎ₂) then ℎ₁⊕ℎ₂ is 𝑜.

With this understanding, it is clear that 𝑡⊕𝜖𝑡𝑤 as defined in Definition 2.2 is the same thing as 𝑡⊕𝜖𝑤⊕(−𝜖)𝑡. Of course, this suggests operations on a vector space.

Remark 2.4. The central role of the derivative of the norm function as featured in Definition 2.2 is not without precedent. The derivative of the norm also plays an important role in semi-inner product spaces [2, 3] (and premanifolds [4]), where the condition that the space be continuous (or, alternatively, uniformly continuous) can be shown to be equivalent to the condition that the norm is Gateaux differentiable (or, alternatively, uniformly Frechet differentiable). Naturally, once the isomorphism between 𝐸 and ℝ² is established, our definition is seen to be analogous to the standard one.

Lemma 2.5. There is an isomorphism between the vector space ℝ² with component-wise addition of elements and component-wise multiplication of elements by scalars, and the Euclidean plane with the Definition 2.2 associated operations.

Proof of Lemma 2.5. We first identify a particular Cartesian axis system on the plane. One Cartesian system is already present, consisting of the lines containing the line segments forming the right angle of the right triangle given at the outset of this proof (i.e., 𝑜𝑎 and 𝑎𝑠). However, since the length function 𝑓(𝑤) is referenced to 𝑜, we use the latter axis system to set up a different Cartesian system at 𝑜. Using the parallel axiom, consider the line though 𝑜 that is parallel to the line containing 𝑎𝑠, and furthermore (again using the parallel axiom) consider a point 𝑏 on this new parallel line such that the vertices {𝑜,𝑎,𝑠,𝑏} form a rectangle. The lines containing 𝑜𝑎 and 𝑜𝑏 are our Cartesian system (the “𝑜𝑎-axis” and the “𝑜𝑏-axis”).
We identify 𝑜 with the ordered pair (0,0). Any point 𝑡 in the plane different from 𝑜 is associated with a unique ordered pair (𝜏_𝑎,𝜏_𝑏) implied by the line segment 𝑜𝑡. That is, we take the orthogonal projection of 𝑡 to the line containing 𝑜𝑎 and take |𝜏_𝑎| to be the length of the line segment formed by 𝑜 and this projection of 𝑡. 𝜏_𝑎 is negative or positive depending on whether or not 𝑜 is between 𝑎 and the projection of 𝑡 to the line containing 𝑜𝑎. 𝜏_𝑏 is defined analogously with respect to 𝑏 and the 𝑜𝑏-axis. Conversely, every ordered pair of real numbers is associated with a point in the plane. That is, for (𝜒_𝑎,𝜒_𝑏) we find the point 𝑥_𝑎 on the 𝑜𝑎-axis with 𝑓(𝑥_𝑎)=|𝜒_𝑎| such that 𝑜 is between 𝑥_𝑎 and 𝑎 if the sign of 𝜒_𝑎 is negative, and 𝑜 is not between 𝑥_𝑎 and 𝑎 if the sign of 𝜒_𝑎 is positive—and similarly for a point 𝑥_𝑏 on the 𝑜𝑏-axis relating to 𝜒_𝑏. Then the point 𝑥 associated with (𝜒_𝑎,𝜒_𝑏) is the point such that {𝑜,𝑥_𝑎,𝑥,𝑥_𝑏} is a rectangle. Its existence and uniqueness is guaranteed by the parallel axiom. It is further obvious that the first construction associates 𝑡 to (𝜏_𝑎,𝜏_𝑏) if and only if the second construction associates (𝜏_𝑎,𝜏_𝑏) to 𝑡.
The above is therefore a one-to-one mapping between the points of the Euclidean plane and the points of ℝ² (ordered pairs of real numbers), explicitly employing the parallel and betweenness axioms. ℝ² becomes a vector space once we specify that ordered pairs (i.e., vectors) representing points in the plane can be added together component-wise and multiplied by scalars component-wise. The isomorphism between vector space ℝ² and 𝐸 with the Definition 2.2 associated operations is then easily verified. Thus, Lemma 2.5 is established.

It follows that the isomorphism in the above lemma leads to an expression for the directional derivative in the vector space ℝ² that gives the same result as it did in the original Euclidean plane. In particular, basic arguments from multivariable calculus establish the existence of a total derivative, the gradient ∇𝑓(𝑠), as the ordered pair of directional derivatives of 𝑓 at 𝑠 with direction line segments defined by unit length line segments parallel to the 𝑜𝑎-axis and 𝑜𝑏-axis such that the directional derivatives are the inner product of ∇𝑓(𝑠) with the ordered pair in ℝ² corresponding to a particular direction line segment. For example, this is seen from 𝑓𝑎₁,𝑏₁−𝑓𝑎₀,𝑏₀=𝑓𝑎₁,𝑏₁−𝑓𝑎₀,𝑏₁+𝑓𝑎₀,𝑏₁−𝑓𝑎₀,𝑏₀=𝜕𝑓𝜉_𝑎𝑎𝜕𝑎¹−𝑎₀+𝜕𝑓𝜉_𝑏𝑏𝜕𝑏¹−𝑏₀, (2.9) with 𝜉_𝑎,𝜉_𝑏 given by the mean value theorem. Applying the definition of directional derivative to both sides above one sees that the directional derivative is given by the usual inner product of ∇𝑓(𝑠)=(𝜕𝑓(𝑠)/𝜕𝑎,𝜕𝑓(𝑠)/𝜕𝑏) with the direction line segment. This is accomplished without use of the Euclidean norm or prior use of the inner product operation. Being an ordered pair, ∇𝑓(𝑠) is a point in the plane, and we can refer to 𝑓(∇𝑓(𝑠)). Furthermore, we have already established that the largest directional derivative of 𝑓 at 𝑠 associated with a line segment 𝑠𝑤 of length 𝜆 is suchthat 𝑜,𝑠,𝑤 are collinear with 𝑠 between 𝑜 and 𝑤. A standard argument establishes that 𝑜, ∇𝑓(𝑠), and 𝑤 are then collinear, and 𝑜 is not between 𝑠 and 𝑤 (because the gradient is proportional to the direction line segment associated with the greatest directional derivative and, according to Lemma 2.3, this direction line segment 𝑠𝑤 lies on the line containing 𝑜𝑠 with 𝑜 not between 𝑠 and 𝑤). So, we have 𝑠=𝛽∇𝑓(𝑠), for 𝛽>0. Furthermore, a standard multivariable calculus argument establishes that the magnitude of ∇𝑓(𝑠) (the length of 𝑜∇𝑓(𝑠), i.e., 𝑓(∇𝑓(𝑠))) is the value of the largest directional derivative associated with a direction line segment of length unity—which according to Lemma 2.3 is unity. Thus, 𝑓(∇𝑓(𝑠))=1 (the magnitude of ∇𝑓(𝑠) is unity), so that 𝑠=𝑓(𝑠)∇𝑓(𝑠) since 𝑓(𝑠) is the length of the hypotenuse 𝑜𝑠.

In fact, it is easy to see that the equations of the last sentence of the prior paragraph pertain not just 𝑠 but to any point 𝑡 in the plane different from 𝑜. They hold trivially if 𝑡 is a point on the 𝑜𝑎-axis or 𝑜𝑏-axis. For any other point 𝑡, we can consider 𝑜𝑡 to be the hypotenuse of a right triangle Δ{𝑜,𝑎_𝑡,𝑡}, where 𝑎_𝑡 is the orthogonal projection of 𝑡 to the 𝑜𝑎-axis, and then proceed in the same manner as we have already done for Δ{𝑜,𝑎,𝑠} (noting that our axes are unchanged). Also, as stated at the outset, 𝑓(𝑜)=0, and lately we have the identification of 𝑜 as the ordered pair (0,0). Including the latter, the equations in the last sentence of the prior paragraph constitute a partial differential equation. For 𝑡 identified in our Cartesian system as (𝜏_𝑎,𝜏_𝑏), it is easily verified by standard differentiation that one solution is 𝑓(𝑡)=𝜏²_𝑎+𝜏²_𝑏. To show that this solution is unique, consider that 𝑡=𝑓(𝑡)∇𝑓(𝑡) means that 𝑡 and ∇𝑓(𝑡) and (0,0) are collinear. Thus, for the points 𝑥 on the line containing 𝑜𝑡, we have the ordinary differential equation 𝑥=𝑓(𝑥)[𝑑𝑓(𝑥)/𝑑𝑥] with 𝑓(𝑑𝑓(𝑥)/𝑑𝑥)=1 and 𝑓(0,0)=0. This has a unique solution, so that the already identified solution, 𝑓(𝑡)=𝜏²_𝑎+𝜏²_𝑏, must be the only solution. Thus, we have derived the Euclidean norm, and hence proved the Pythagorean theorem.

Equations (1.3) and (1.4) could also be used in the definition of the Euclidean norm as follows.

Corollary 2.6. A function 𝑓∶ℝ^𝑛→ℝ is the Euclidean norm if and only if it is continuous, vanishes at the origin, and at any other point it satisfies (1.3) and (1.4).

Equations (1.3) and (1.4) indicate that, from a differentiable viewpoint, the Euclidean norm is a scaling-orientation function in the decomposition of a point as the product of a scalar with a unity-scaled orientation point derived from the function's gradient. We can consider this set of equations to represent “Euclidean decomposition."

But the Euclidean norm does not address the multiplicative structure of an algebra and so does not have an essential role in most algebras. Instead, we shall see that the role of ∇𝑓(𝑠) on the vector space ℝ^𝑛 is taken up by ∇^∗𝑓(𝑠⁻¹) on topological *-algebras over ℝ^𝑛, and we will consider (1.3) and (1.4) so modified to represent “Jacobian decomposition.”

3. Jacobian Decomposition and Inverse Norm

The “defining” equations of the Euclidean norm, (1.3) and (1.4), make no reference to the multiplicative structure of an algebra. Nevertheless, the Euclidean norm has application to the Cayley-Dickson algebras (an unending sequence of real unital topological *-algebras beginning with the only four real normed division algebras, ℝ, ℂ, the quaternions, and the octonions). Each of these algebras is characterized by a basis {𝑒₀,…,𝑒_𝑚}, with 𝑚=2^𝑘−1 for any nonnegative integer 𝑘. Multiplication is distributive and thus defined by a multiplication table relevant to {𝑒_𝑖}. In particular, 𝑒²₀=1 and 𝑒²_𝑖=−1 for 𝑖≠0. Any point 𝑠=𝛼₀𝑒₀+⋯+𝛼_𝑚𝑒_𝑚 (with each 𝛼_𝑖∈ℝ) has a conjugate 𝑠^∗≡𝛼₀−⋯−𝛼_𝑚𝑒_𝑚, with ∗ evidently an involution. In particular, 𝑠𝑠^∗=𝛼²₀+⋯+𝛼²_𝑚=[𝑓(𝑠)]², where 𝑓(𝑠) is the Euclidean norm. Thus, 𝑠⁻¹=𝑠^∗/[𝑓(𝑠)]². Hence, the Euclidean norm in this case helps express the inverse of a point. Being the Euclidean norm, 𝑓(𝑠) satisfies (1.3) and (1.4), and so also defines a Euclidean decomposition of the point. But given the above involution, it is easy to show that ∇𝑓(𝑠)=∇^∗𝑓𝑠⁻¹, (3.1) (where, as always, ∇^∗𝑓(𝑠⁻¹) represents evaluation of the gradient ∇𝑓 at the point 𝑠⁻¹ followed by application of the involution). Substituting the above into (1.3), we obtain 𝑠=𝑓(𝑠)∇^∗𝑓(𝑠⁻¹). Since (1.4) holds for all 𝑠≠0, and all such points have inverses, we must also have 𝑓(∇^∗𝑓(𝑠⁻¹))=1. Now we have a formulation for the Euclidean norm that makes reference to unital algebraic structure. On Cayley-Dickson algebras, it is equivalent to the Pythagorean theorem. When this latter decomposition exists on a topological ∗-algebra but 𝑓(𝑠) is not the Euclidean norm, it can be considered to be an algebraic ghost of the Pythagorean theorem.

Definition 3.1. For a topological ∗-algebra 𝐴 defined on ℝ^𝑛, a continuous function 𝑓∶ℂ^𝑛→ℂ is an inverse norm if it is zero on the nonunits of 𝐴, as a function restricted to the domain ℝ^𝑛 it is differentiable on the units of 𝐴, and for any unit, 𝑠=𝑓(𝑠)∇^∗𝑓𝑠⁻¹, (3.2) with 𝑓∇^∗𝑓𝑠⁻¹=1. (3.3)

The above equations mimic the equations for the Euclidean norm referred to in Corollary 2.6, but instead decompose a point as a function's value at the point multiplied by a unity-scaled orientation point dependent on the function's gradient at the inverse of the point (or, alternatively to the Euclidean norm's direct expression of a point, (3.2) and (3.3) express the inverse of a point—i.e., substituting 𝑠⁻¹ for 𝑠 in the latter equations). Thus, we use the term “inverse norm.”

Of course, from (3.1) we have already shown the following.

Theorem 3.2. The Euclidean norm is an inverse norm on the Cayley-Dickson algebras.

However, inverse norms have applicability well beyond the Cayley-Dickson algebras.

Theorem 3.3 (Jacobi). For s a member of the algebra of real matrices ℝ^𝑛×𝑛, let 𝑓(𝑠)≡det(𝑠). If 𝑠 is a unit, then 𝑠=𝑓(𝑠)∇^∗𝑓𝑠⁻¹, (3.4) where * indicates matrix transpose.

The above is a well-known immediate consequence of the Jacobi’s formula in matrix calculus (the latter expresses gradient of the determinant in terms of the adjugate matrix).

Corollary 3.4. For the algebra of real matrices ℝ^𝑛×𝑛, 𝑓(𝑠)≡√𝑛(det(𝑠))^1/𝑛 (3.5) is an inverse norm.

Proof. 𝑓(𝑠) is evidently continuous everywhere, as well as differentiable on the units (the invertible matrices), and it vanishes on the nonunits. For any unit 𝑠 on this algebra, we have det(𝑠)≠0, and ∇^∗𝑓(𝑠)=𝜕𝑓(𝑠)𝜕𝑠_𝑖𝑗^∗=√𝑛𝑛(det(𝑠))^−1+1/𝑛(∇(det(𝑠)))^∗. (3.6) For a unit 𝑠, (3.4) implies (∇det(𝑠))^∗=𝑠⁻¹det(𝑠). Substituting this into (3.6), and using (3.5), we obtain ∇^∗𝑓(𝑠)=𝑓(𝑠)𝑛𝑠⁻¹=𝑠⁻¹𝑓𝑠⁻¹. (3.7) If we evaluate ∇^∗𝑓 on the left-hand-side above at the point 𝑠⁻¹ instead of evaluating it at 𝑠, we obtain (3.2). Applying 𝑓 in (3.5) to both sides of (3.7) we obtain (3.3).

In analogy with Euclidean decomposition (1.3) and (1.4), we can consider the equations of Definition 3.1 to represent “Jacobian decomposition” (i.e., in view of Theorem 3.3).

For the algebra ℝ^𝑛 with component-wise addition and multiplication, it is also easy to show that an inverse norm is given by the product of √𝑛 with the geometric mean of the absolute values of the components of a point. That is, for a point 𝑠=(𝑠₁,…,𝑠_𝑛), set 𝑓(𝑠)≡√𝑛^𝑛_𝑖=1𝑠_𝑖^1/𝑛. (3.8) Note that 𝑓 is continuous and 𝑓 vanishes on the nonunits. If 𝑠 is a unit, then 𝜕𝑓(𝑠)𝜕𝑠_𝑗=√𝑛𝑛^𝑛_𝑖=1𝑠_𝑖^−1+1/𝑛sgn𝑠_{𝑗𝑖≠𝑗}𝑠_𝑖=√𝑛𝑛^𝑛_𝑖=1𝑠_𝑖^1/𝑛1𝑠_𝑗=𝑓(𝑠)𝑛1𝑠_𝑗. (3.9) It is then a simple task to verify that 𝑓(𝑠) satisfies the requirements of Definition 3.1 (with∗ as the identity).

Now we turn to Jordan algebras.

Theorem 3.5. The Minkowski Norm is an inverse norm on the spin factor jordan algebra.

Proof. Thinking of ℝ^𝑛 in the format of ℝ⊕ℝ^𝑛−1, write its points as 𝑠=𝑤+𝐳, with 𝑤∈ℝ and 𝐳=(𝑧₁,…,𝑧_𝑛−1)∈ℝ^𝑛−1. We introduce a multiplication operation such that 𝜔_𝑎+𝐳₁𝜔_𝑏+𝐳₂≡𝜔_𝑎𝜔_𝑏+𝐳₁⋅𝐳₂+𝜔_𝑎𝐳₂+𝜔_𝑏𝐳₁, (3.10) where “⋅” is the usual inner product on ℝ^𝑛−1. This multiplication defines a commutative but nonassociative algebra, the spin factor Jordan algebra [5]. The multiplicative identity element is evidently the point where 𝑤=1 and 𝐳=0. An inverse element exists for points 𝑤+𝐳 such that 𝐳⋅𝐳≠𝑤². That is, (𝑤+𝐳)(𝑤+𝐳)⁻¹=1 for (𝑤+𝐳)⁻¹=−𝑤+𝐳−𝑤²,+𝐳⋅𝐳 (3.11) when 𝐳⋅𝐳≠𝑤².
Now we define 𝑓∶ℂ^𝑛→ℂ such that for 𝑠=(𝑤,𝑧₁,…,𝑧_𝑛−1)∈ℂ^𝑛, 𝑓(𝑠)=−𝑤²+𝑧²₁+⋯+𝑧²_𝑛−1, (3.12) where the above square root represents the principal value. On the domain comprised of the units of the spin factor Jordan algebra (the points 𝑤+𝐳, 𝑤∈ℝ,𝐳∈ℝ³, such that 𝑤²≠𝐳⋅𝐳), we have ∇𝑓(𝑤+𝐳)=−𝑤+𝐳𝑓()=()𝑤+𝐳𝑤+𝐳⁻¹𝑓()=()𝑤+𝐳𝑤+𝐳⁻¹𝑓(𝑤+𝐳)⁻¹, (3.13) where the three equalities follow from (3.11) and (3.12). Hence, on the units, we have 𝑠⁻¹=𝑓(𝑠⁻¹)∇𝑓(𝑠). Taking the involution ∗ to be the identity, the latter equation is equivalent to the first equation of Definition 3.1.
Applying 𝑓 to both sides of the first equality in (3.13) and using (3.12), we obtain 𝑓(∇𝑓(𝑠))=1 for any unit 𝑠. This is equivalent to the second equation of Definition 3.1.

On the other hand, the Jordan algebra obtained from the algebra of matrices ℝ^𝑛×𝑛 has the product of two of its members 𝐴,𝐵 as given by 𝐴⋆𝐵≡(𝐴𝐵+𝐵𝐴)/2 where 𝐴𝐵 and 𝐵𝐴 indicate the usual matrix product. We then have 𝟏=𝐴⋆𝐴⁻¹=𝐴𝐴⁻¹, where 1 is the identity element in the algebra (in this case, 𝟏=diag{1,1,…,1}). Therefore, 𝐴⁻¹ is the usual matrix inverse. Consequently, the associated inverse norms are the same as those for the algebra of matrices ℝ^𝑛×𝑛. Thus, Jacobian decomposition holds for the Jordan algebra obtained from the matrix algebra.

Supplying an inverse norm nominally requires solution of a nonlinear partial differential equation (3.2). However, if we apply further restrictions on the nature of 𝑓(𝑠), one can obtain a linear equation. In particular, for each algebra example considered up till now there is a constant 𝛼∈ℝ such that 𝑓(𝑠)𝑓𝑠⁻¹=𝛼, (3.14) so that (3.2) evaluated at 𝑠⁻¹ instead of 𝑠 implies (1.6) and 𝛼𝑠∇^∗𝑓(𝑠)=𝑓(𝑠)𝟏. (3.15)

However, not all unital algebras have an inverse norm satisfying (3.15). First, since an inverse norm is zero on nonunits, and a nonunital algebra consists only of nonunits, the inverse norm on a nonunital algebra is identically zero. On the other hand, one might ask whether an inverse norm satisfying (3.15) exists on the unital hull [5] of a nonunital topological algebra.

Theorem 3.6. The unital hull of a nonunital topological algebra does not have an inverse norm satisfying (3.15).

Proof. The unital hull of a nonunital algebra 𝐴 is defined by elements 𝑠̂≡(𝜎,𝑠) for 𝜎∈ℝ and 𝑠=(𝑠₁,…,𝑠_𝑛)∈𝐴, with component-wise addition of elements and multiplication defined by (𝜎,𝑠)⋆(𝜏,𝑡)≡(𝜎𝜏,𝜎𝑡+𝜏𝑠+𝑠𝑡), (3.16) where 𝑠𝑡 indicates the product between elements of 𝐴. The identity element is 𝟏=(1,0). Equation (3.15) requires that the units satisfy ()⋆()𝜎,𝑠𝜕𝑓𝑠̂()=𝑓()𝜕𝜎,∇𝑓𝑠̂𝑠̂𝛼𝟏, (3.17) where ∇≡(𝜕/𝜕𝑠₁,…,𝜕/𝜕𝑠_𝑛). Using the multiplication rule, we can write this as 𝜎𝜕𝑓(𝑠)̂()()𝜕𝜎,𝜎∇𝑓𝑠̂+𝑠𝜕𝑓𝑠̂()=𝑓()𝜕𝜎+𝑠∇𝑓𝑠̂𝑠̂.𝛼,0 (3.18) Thus, it is required that 𝜎[𝜕𝑓(𝑠)̂/𝜕𝜎]=𝑓(𝑠)̂/𝛼, so that 𝑓(𝑠)̂=𝜎^1/𝛼ℎ(𝑠), where ℎ(𝑠) indicates some function independent of 𝜎. In addition, the right-hand-side of (3.18) requires that the second component of the left-hand-side of (3.18) be zero. But with regard to variation in 𝜎, the requirement that 𝑓(𝑠)̂=𝜎^1/𝛼ℎ(𝑠) means that the first term of this second component is 𝑂(𝜎^1+1/𝛼), the second term is 𝑂(𝜎^−1+1/𝛼), and the third term is 𝑂(𝜎^1/𝛼). It is thus impossible for this second component to remain zero as 𝜎 varies unless 𝑓 is identically zero. But in that case, the equations of Definition 3.1 cannot be satisfied for the units. Thus, an inverse norm satisfying (3.15) does not exist.

One can get even more restrictive and consider algebras for which not only is 𝑓(𝑠)𝑓(𝑠⁻¹) constant on the units (i.e., (3.14) is satisfied) but in addition the units constitute a group on which a multiple of 𝑓(𝑠) is homomorphism. In fact, the inverse norm on the first four Cayley-Dickson algebras satisfies this prescription (the latter are the only real normed division algebras by Hurwitz's theorem, and the Euclidean norm is a homomorphism on their units). The inverse norm 𝑓(𝑠)=√𝑛(det(𝑠))^1/𝑛 on the matrix algebra ℝ^𝑛×𝑛 also satisfies this requirement (i.e., 𝑓(𝑠)/√𝑛 is a homomorphism on the group of units). Along these lines, we observe that the Cayley-Dickson algebras and the set of subalgebras of the real matrix algebra ℝ^𝑛×𝑛 overlap on the algebras of real numbers, the complex numbers, and quaternions—which happen to be the only associative real normed division algebras. The Cayley-Dickson algebra sequence contains the only other real normed division algebra—the (noncommutative/nonassociative) algebra of octonions. Thus, the octonions provide an example of an algebra with an inverse norm that is a homomorphism on the group of units, but without a representation as a matrix subalgebra. This point of nonoverlap is typical of the exceptionalism of the octonions [6].