*This is a series of notes taken during my review of linear algebra, using Axler's excellent textbook Linear Algera Done Right, which will be heavily referenced.*

Every complex number can be written as a nonnegative real number \(r\) times a number \(e^{i \theta}\) on the unit circle:

Similarly, every operator \(T\) can be written as an isometry \(S\) times a positive semidefinite (positive definite when \(T\) is invertible) operator:

This theorem has a nice physical manifestation: in continuum mechanics, any deformation can be separated into a pure rotation effect and pure compression effect (as an imprecise analogy, a Rasengan should to hurled at the opponent with the maximum rotation and velocity in order to deliver the greatest damage).

Example, exercise 7.D.7: Define \(T \in \mathcal{L}(\mathbb{F}^3)\) by \(T(z_1, z_2, z_3) = (z_3,2z_1,3z_2)\). Find (explicitly) an isometry \(S \in \mathcal{L}(\mathbb{F}^3)\) such that \(T = S \sqrt{T^*T}\).

Solution: with respect to the standard basis,

By inspection, \(\mathcal{M}(\sqrt{T^*T})\) differs from \(\mathcal{M}(T)\) only in the order of the entries, so if we define \(S\) by \(S(z_1,z_2,z_3) = (z_3,z_1,z_2)\), then \(S\) is certainly an isometry, and we arrive at the beautiful decomposition:

We have decomposed the matrix description of \(T\) into a pure rotation matrix \(\mathcal{M}(S)\) and a (positive) "stretching" matrix \(\mathcal{M}(\sqrt{T^*T})\)