Linear Algebra and Its Applications, Exercise 3.4.4

Exercise 3.4.4. Given two orthogonal matrices $Q_1$ and $Q_2$, show that their product $Q_1Q_2$ is also orthogonal. If $Q_1$ represents rotation through the angle $\theta$ and $Q_1$ represents rotation through the angle $\phi$, what does $Q_1Q_2$ represent? What trigonometric identities for $\sin (\theta + \phi)$ and $\cos (\theta + \phi)$ can be found in multiplying $Q_1$ and $Q_2$?

Answer: For any orthogonal matrix $Q$ we have $Q^TQ = I$. We then have

$(Q_1Q_2)^T(Q_1Q2) = (Q_2^TQ_1^T)(Q_1Q2) = Q_2^T(Q_1^TQ_1)Q_2 = Q_2^TQ_2 = I$

Since $(Q_1Q_2)^T(Q_1Q2) = I$ the matrix $Q_1Q_2$ is orthogonal.

If the matrices $Q_1$ and $Q_2$ represent rotations through $\theta$ and $\phi$ respectively, then $Q_1Q_2$ represents rotation through $\theta + \phi$, and will contain elements including $\sin (\theta + \phi)$ and $\cos (\theta + \phi)$. These will be produced in the matrix multiplication through the following trigonometric identities:

$\sin (\theta + \phi) = \sin \theta \cos \phi + \cos \theta \sin \phi$

$\cos (\theta + \phi) = \cos \theta \cos \phi - \sin \theta \sin \phi$

where $\sin \theta$ and $\cos \theta$ come from $Q_1$  and $\sin \phi$ and $\cos \phi$ come from $Q_2$.

NOTE: This continues a series of posts containing worked out exercises from the (out of print) book Linear Algebra and Its Applications, Third Edition by Gilbert Strang.

If you find these posts useful I encourage you to also check out the more current Linear Algebra and Its Applications, Fourth Edition, Dr Strang’s introductory textbook Introduction to Linear Algebra, Fifth Edition and the accompanying free online course, and Dr Strang’s other books.

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