Prove: T is Angle-Preserving iff |ai| are Equal

[Adeimantus]

Homework Statement

1. Problem 1-8(b)

If there is a basis x₁,..., x_n of Rⁿ and numbers a₁,..., a_n such that Tx_i = a_ix_i (where T is a linear operator), prove that T is angle-preserving if and only if all the |a_i| are equal.

Homework Equations

First a couple of definitions...

Definition (i): If x,y are nonzero vectors in Rⁿ, the angle between them is given by

angle(x,y) = arccos {<x,y>/ ( |x | |y | )}

where < , > denotes the standard inner product and | | the standard norm.

Definition (ii): A linear transformation T is angle-preserving if T is 1-1 and for x,y != 0 we have that angle(Tx, Ty) = angle(x,y).

The Attempt at a Solution

Problem is, I think I see a simple counterexample to the sufficiency part of the proof. Let x₁, x₂ be a basis for R² where

x₁ = e₁ (i.e. standard unit basis vector) and
x₂ = e₁ + e₂

Then suppose Tx₁ = x₁ and Tx₂ = -x₂. Now consider the angle between x₁ and x₁+x₂:

angle(x₁, x₁+x₂) = angle(e₁, 2e₁+e₂) ~ 25 degrees, but

angle(Tx₁,T(x₁+x₂)) = angle(x₁, x₁-x₂) = angle(e₁, -e₂) = 90 degrees.

So even though the absolute values of a₁ = 1 and a₂ = -1 are equal, it seems T does not preserve angles. What am I misunderstanding about this problem? I am assuming the author means angle-preserving with respect to the standard inner product in which <e₁, e₂> = 0.

 

[poisson-aerohead]

Hi, I know this is a blast from the distant past (14.5 years?) but I was just doing this problem and ran into this exact issue. I, in fact, constructed the identical counter example that you did. When I searched Spivak problem 1-8 and found this, I had to make an account just to say something. IMO, your conclusion is correct, the $|\lambda|$ condition is only for orthogonal bases.

Did you ever get further confirmation on this? I am going to go forward under the assumption that the condition is in general equal lambda except for orthogonal bases, where it expands to equal absolute value of lambda. At any rate, I like this book but Spivak can be a bit fast and loose sometimes, and this seems to be one of those examples where he could stand for a bit more rigorous language at the expense of making the book a bit longer. The book is short so a few more pages would not be too bad, and IMO make the book easier (since I often find myself filling in what he means).

FWIW, I would bet the technical reason behind this outcome is basically that $0 = -0$. When you equate the cosine subject to the condition that each transformed eigenvector remains co-linear with itself, there are three options (from a 2D perspective). You can leave both fixed (which does not change the relative angle, or any angles), advance both 180 degrees (which does not change the relative angle), or advance one by 180 degrees (which does change the relative angle and corresponds to the lambdas being of opposite sign). In this case, you need to solve:
$$
\begin{align*}
\cos(\theta) &= \cos(\theta+\pi) \\
\cos(\theta) &= \cos(\theta)\cos(\pi) - \sin(\theta)\sin(\pi) \\
\cos(\theta) &= - \cos(\theta)
\end{align*}
$$
This is clearly only possible if $\cos(\theta) = 0$. In higher dimensions, the above analysis is true for the 2D subspace containing the two eigenvectors.

 

[WWGD]

Homework Statement

1. Problem 1-8(b)

If there is a basis x₁,..., x_n of Rⁿ and numbers a₁,..., a_n such that Tx_i = a_ix_i (where T is a linear operator), prove that T is angle-preserving if and only if all the |a_i| are equal.

Homework Equations

First a couple of definitions...

Definition (i): If x,y are nonzero vectors in Rⁿ, the angle between them is given by

angle(x,y) = arccos {<x,y>/ ( |x | |y | )}

where < , > denotes the standard inner product and | | the standard norm.

Definition (ii): A linear transformation T is angle-preserving if T is 1-1 and for x,y != 0 we have that angle(Tx, Ty) = angle(x,y).

The Attempt at a Solution

Problem is, I think I see a simple counterexample to the sufficiency part of the proof. Let x₁, x₂ be a basis for R² where

x₁ = e₁ (i.e. standard unit basis vector) and
x₂ = e₁ + e₂

Then suppose Tx₁ = x₁ and Tx₂ = -x₂. Now consider the angle between x₁ and x₁+x₂:

angle(x₁, x₁+x₂) = angle(e₁, 2e₁+e₂) ~ 25 degrees, but

angle(Tx₁,T(x₁+x₂)) = angle(x₁, x₁-x₂) = angle(e₁, -e₂) = 90 degrees.

So even though the absolute values of a₁ = 1 and a₂ = -1 are equal, it seems T does not preserve angles. What am I misunderstanding about this problem? I am assuming the author means angle-preserving with respect to the standard inner product in which <e₁, e₂> = 0.

Maybe you can use that the only matrices that preserve angles are the orthogonal matrices to craft ant
(counter) argument.