Linear Algebra and Its Applications · Problem 3.4.4

Orthogonality — Gilbert Strang

Mathematics · orthogonal-matrix-product

Problem

Q_1

and

Q_2

are orthogonal matrices, so that

Q^T Q = I

, show that

Q_1 Q_2

is also orthogonal. If

Q_1

is rotation through

\theta

, and

Q_2

is rotation through

\phi

, what is

Q_1 Q_2

? Can you find the trigonometric identities for

\sin(\theta + \phi)

and

\cos(\theta + \phi)

in the matrix multiplication

Q_1 Q_2