Lean, informalized

Nielsen & Chuang · Chapter 8, Quantum noise and quantum operations · p. 375

Exercise 8.14 - A proper rotation times a symmetric matrix whose determinant is negative

A rotation times a symmetric factor of negative determinant.

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The Exercise

A real 3×33\times 3 matrix MM can be written as a product M=OSM = OS of a proper rotation OO and a real symmetric matrix SS. The symmetric factor stretches space along three perpendicular axes, and the rotation then turns the result. The exercise is to show that SS may have a negative determinant, so that besides stretching it reflects space.

Rotations and Symmetric Matrices

The special orthogonal group SO(3)\mathrm{SO}(3) is the group of proper rotations of space: the real 3×33\times 3 matrices OO with OTO=I,det⁡O=1.O^{T} O = I, \qquad \det O = 1.(1) A real 3×33\times 3 matrix SS is symmetric when it equals its transpose, ST=SS^{T} = S.

The Claim

Theorem 1. There are real 3×33\times 3 matrices MM, OO and SS, with OO a proper rotation in SO(3)\mathrm{SO}(3) and SS symmetric, such that M=OS,det⁡S<0.M = OS, \qquad \det S < 0.(2)✦

What Is Checked

The claim is stated over real 3×33\times 3 matrices, with the group SO(3)\mathrm{SO}(3) of (1) and the symmetric matrices.