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I've been curious which common classes of linear transformations correspond to real vs imaginary eigenvalues. For example, matrices which cause rotation in a linear transformation have imaginary eigenvalues.

Edit: Let's distinguish real from complex (as opposed to purely imaginary; references Ben Grossmann's comment)

Which others is this true for, specifically from the Wiki list?

  • Stretching
  • Squeezing
  • Shearing
  • Reflection
  • Orthogonal Projection

The following Q/A were a bit to proofy for me to follow. So the scope of this question is prioritizes "what" over "why." (Though, I'd happily accept both.)

Why do complex eigenvalues correspond to a rotation of the vector?

Imaginary eigenvalues

jbuddy_13
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    By "imaginary" do you mean "purely imaginary" or "complex"? For example, it's clear that $3i$ is an imaginary number, but do you consider $2 + 3i$ to be imaginary? – Ben Grossmann Oct 06 '21 at 16:49
  • @BenGrossmann, great question! Let's distinguish real vs complex, as opposed to purely imaginary – jbuddy_13 Oct 06 '21 at 16:50
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    None of those five have imaginary eigenvalues. The action of a pair of pure imaginary eigenvalues for a real matrix in real space is rotation in a 2D plane, that's all. – Ian Oct 06 '21 at 16:56
  • @Ian, Thanks! Though, I'm curious why rotation in 2D has imaginary EV but reflection does not. Any hints there? – jbuddy_13 Oct 06 '21 at 17:01
  • A pure reflection would be something like $\begin{bmatrix} 1 & 0 \ 0 & -1 \end{bmatrix}$. It sends vectors in the "mirror" to themselves and vectors perpendicular to the "mirror" to their negative. – Ian Oct 06 '21 at 17:08
  • @jbuddy_13 Although I think it's perfectly clear what you mean, my inner pedant (as well as my outer pedant) wants to remind you that real numbers are actually complex numbers too. So, you want to distinguish between real numbers and non-real complex numbers. :-) – Theo Bendit Oct 06 '21 at 17:21

1 Answers1

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To go through the list:

  • Every stretching/squeezing transformation has real, non-negative eigenvalues
  • Every shear transformation has $1$ as its only eigenvalue
  • Every reflection has eigenvalues $1$ or $-1$ (typically both)
  • Every orthogonal projection has eigenvalues $0$ or $1$ (typically both)

As you noted, rotation matrices will have non-real eigenvalues, except in the case that the rotation angle(s) is a multiple of $180^\circ$. In other words, the eigenvalues will be non-real except in the case that the transformation can be thought of as a combination of "mutually orthogonal" reflections. Similarly, orthogonal transformations (transformations that can be done by a combination of reflections and rotations) will have non-real eigenvalues except in the case that all eigenvalues can be accounted for by a set of independent reflections.

Notably, reflections, orthogonal projections, and rotations (along with the orthogonal transformations) all fall into the class of "normal" transformation. A transformation is called normal if its matrix $A$ satisfies $A^TA = AA^T$ (where $A^T$ denotes the transpose of $A$). It turns out that a normal transformation will have real eigenvalues if and only if it is symmetric (that is, $A = A^T$).

Symmetric transformations all correspond to simultaneous streches/squeezes/reflections along mutually orthogonal axes. Normal transformations generalize this in that they also allow these transformations to include rotations about these same orthogonal axes (or more generally, within the planes containing any pair of axes).

Ben Grossmann
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