Reciprocal Eigenvalues

The title of this course uses two technical terms and draws attention from people familiar with them.

The reciprocation of a number, to produce a multiplicative inverse, is an algebraic operation that is singular at zero.

Eigenvalues are properties of certain matrices in linear algebra. They are associated with eigenvectors v. If matrix T operates on a row vector v to produce v T = a v, then the number a is an eigenvalue for T. It means that for a line {x v : x in R} in a vector space, T acts as a magnification if a > 1, and as a contraction if 0 < a < 1. The negative a cases mean that T reflects the line through the origin (zero vector).

In this course two dimensions suffice, so there can be two eigenvalues, in this case reciprocals of one another. Then T can be written as a diagonal matrix $$\begin{pmatrix}a & 0 \\ 0 & 1/a \end{pmatrix} .$$

For example, (1, 1) T = (a, 1/a). At the origin there is a square at (1,1) and a rectangle at (a, 1/a). The rectangle, having length and width as reciprocals, has the same area as does the square. In a perfectly elastic plane, the operation of T can be called a squeeze of parameter a.

Stable level curves
Given any constant c > 0, there is a hyperbola $$H(c)=\{(x,y) : xy = c \} .$$

The application of a squeeze, of whatever parameter a, to H(c) leaves the hyperbola stable:
 * (x,y) in H(c) implies (a x, y/a) in H(c).

Given any c > 0, H(c) can be called a level curve of parameter c. Use Q to represent the quadrant with x > 0, y > 0. For any c, the region contained by the asymptotes and H(c) is stable under squeezes.

Now take c = 1, call H(1) the standard hyperbola, and consider the region it bounds with the asymptotes. A subset of the region is the descending staircase of steps of height y = 1/n over the interval [n&minus;1, n]. The sum of the areas under the stairs is called "the harmonic series". A student must learn to show that this area is unbounded.

When the squeeze parameter is taken as a variable, its various actions on Q can be viewed with each H(c) as a streamline in a corner flow. With a > 1 the flow descends and veers right. With 0 < a < 1 the flow reverses.

Two points on an H(c) and the radial lines to them determine a hyperbolic sector. Such a sector is mapped to another sector of equal area by a squeeze. One might ask, fpr the standard hyperbola, what x makes the sector between (1,1) and (x, 1/x) have unit area ? Leonard Euler found the answer to be e = 2.718281828 approximately. The number cannot be expressed as a rational fraction of integers, nor as a solution to an algebraic equation (It is called "transcendental").

Though area is preserved by a squeeze, shape is distorted and Euclidean distances changed under squeezing. For example, a sector near (1,1) has a broader shape than its image when a >> 1. These transformed sectors are so narrow that they appear as lines in Q.

To standardize area measure of sectors, a sector of one unit is one wing. The rays defining a sector can be viewed as a hyperbolic angle. For Euler number e, the angle between (1,1) and (e, 1/e) has area equal to one wing. Squeezing now with a = e, the image of the above sector is between (e, 1/e) and (e2, 1/ e2), which has another wing of area. As every pair (en, 1/ en) and (en+1, 1/ en+1) contributes a wing to the total area, there is no upper bound on the measure of the area of a hyperbolic sector or of the size of a hyperbolic angle.

Proponents of angle as area
The notion of angle size being related to area measurement has had proponents and detractors. As has been shown, hyperbolic angle depends on area for its definition, but what of circular angle? In the fourth century Theon of Alexandria wrote "the area of sectors of a circle are proportional to their angles at the center."

The idea of unifying the circular and hyperbolic angle by reference to sector areas was propounded by Robert Baldwin Hayward in 1892. The following year Alexander Macfarlane proposed this unification in a paper "On the definition of the trigonometric functions" which he submitted for consideration at the mathematical conference held in Chicago in connection with the Columbia Exposition. For some reason the paper was withdrawn and published later in his Essays on Space Analysis.

An exposition of the unification of angles through the notion of sector areas has been contributed to the Wikibook Geometry in the chapter Unified Angles.

Dented trapezoids
Consider a hyperbolic sector that extends from (1,1) to (a, 1/a). To develop the calculus of a single variable, another view of this sector area is taken by addition and subtraction of triangles of area one-half. First consider the right triangle with base [0, a] and altitude 1/a. Join this triangle to the sector, then take away the right triangle on base [0,1] of altitude 1. The remaining region has base [1, a], parallel sides at x=1 and x=a, and a concave top determined by y=1/x. This region will be called a dented trapezoid, and its area is equal to the area of the hyperbolic sector. For students of calculus this area is familiar as an expression of the natural logarithm of a. Except for the Wikibook Calculus, the integration of the function f(x) = 1/x over an interval [1, a] is introduced without mention of hyperbolic sectors. Evidently the steps given above for approaching the dented trapezoid have been obviated by other calculus texts. Nevertheless, the area of the sector is log a. When 0 < a < 1, the logarithm is negative, so evidently the area of the sector between [a, 1/a] and [1, 1] is taken as negative area.

Lemma: The area of a dented trapezoid over [a,b] depends only on the ratio b/a.
 * proof: Squeeze mapping with parameter c moves the sector determined by [a,b] to the sector of [ca, cb].

Theorem: Log ab = log a + log b.
 * proof: Log a + log b represents the sum of the areas of trapezoids over [1,a] and [1,b]. By the lemma, the area over [1,b] equals the area over [a, ab]. Joining the trapezoids at x=a yields one over [1, ab] which represents log ab.

The measure of dented trapezoids over [a,1] where 0 < a < 1 follows from the symmetry of the standard hyperbola with respect to the line y = x. A reflection of a sector to (a, 1/a) produces the sector to (1/a, a) which is considered negative. Algebraically, ab = 1 means log ab = 0 since the dented trapezoid at 1 collapses to a segment which has measure zero. Thus log (1/a) = &minus; log a.

Corollary: Log (b/a) = log b &minus; log a.

Meter &sdot; second
The meter&sdot;second has been used in kinematics on a cartesian spacetime plane, where velocity v is the aspect ratio of a rectangle in a frame of reference. A moving observer on the diagonal of this rectangle establishes a rectangle in his own frame which appears as a parallelogram in the reference frame. The areas of the rectangle and parallelogram agree, and the transformation of the plane is called a shear mapping, acting on row vectors through:
 * $$(t,\ x) \begin{pmatrix}1 & v \\ 0 & 1 \end{pmatrix} \ = \ (t,\ x\ + tv).$$

Kinematics in this framework prevailed until the group of shears $$\{ \begin{pmatrix}1 & v \\ 0 & 1 \end{pmatrix}\ : v \in \reals \} $$ was observed to be unbounded. But physical velocities are bounded by the speed of light, so a change of transformation groups was made for modern kinematics. The composition of velocities in the new group is bounded by light speed.

H. A. Lorentz wrote some transformation equations that replaced the old kinematics. These transformations in a spacetime plane are squeeze mappings. They share the shear property of area-preservation while altering distances. Instead of the quadratic form $$t^2 + x^2$$ they respect $$t^2 - x^2 \ .$$ The flat cosmology introduced in 1908 by Hermann Minkowski was not an innovative fusion of spacetime as he claimed, but the cosmological picture of Galileo and Newton was replaced.

Beam space


In a spacetime plane there is only one dimension of physical space, two dimensions are missing. Let $$L = \{ (0,\ x) : x \in \Reals \}$$ be the line at a temporal origin. Now light beams may follow this line, one to the right, the other left. Conventionally, time is graphed vertically, space horizontally, so (t, x) are read with vertical first, horizontal second, contrary to usual order of coordinates.

In order to supply a symmetric figure, some units must be introduced for speed of light: one English foot is crossed in a nanosecond = 10&minus;9 s. On the planetary scale, the distance to the sun and back is crossed in about 103 seconds. For intergalactic distances, it takes an Earth-year for light to travel a "light year". With these units the beams from (0,0) left and right follow 45° and 135° lines. The cross of these lines separate the future, space left, the past, and space right. The squeeze mapping is applied to the future quadrant, and to the past by symmetry. The space quadrants are also squeezed by reflecting the future motion in a diagonal. Thus all four quadrants are transformed; there is no mingling of the quadrants.

A speed of 1.5 mile-per-hour corresponds to a nanowing of squeeze. So a fastball from a baseball pitcher's mound may clock out at 150 nanowings.

Poise
The unit poise is given as kilograms per meter&sdot;second, and is a measure of dynamic viscosity &mu;. The ratio of &mu; to density is called kinematic viscosity since gravity, or other massive forces, have been cancelled by the ratio.

Osborne Reynolds is remembered for his experiments providing an estimate for the onset of turbulence out of a laminar flow. The viscosity keeps material particles in order in a flow, and the disorder occurs with a high ratio of density times flow speed times a length characteristic of the region to &mu;. This ratio is called Reynolds number Re.

Readings

 * Abstract Algebra/Shear and Slope
 * Kinematics/Transformations
 * Calculus/Hyperbolic logarithm and angles
 * Mathematics for Economics/Hyperbolas