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

Sector between (a, 1/a) and (b, 1/b)

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.

A subset of the region is the descending staircase of steps of height y = 1/n over the interval [n−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, what x makes the sector between (1,1) and (x, 1/x) have unit area ? w: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.

Readings

Completion status: this resource is ~25% complete.