Definition 1. A square matrix is called orthogonal if
Exercise 1. Let be orthogonal. Then a) b) the transpose is orthogonal, c) the inverse is orthogonal, d)
Proof. a) is the left inverse of Hence, is invertible and its inverse is b) from the inverse definition. Part c) follows from parts a) and b). d) Just apply to the definition to get
Exercise 2. An orthogonal matrix preserves scalar products, norms and angles.
Proof. For any vectors scalar products are preserved: Therefore vector lengths are preserved: Cosines of angles are preserved too, because Thus angles are preserved.
Since the origin is unchanged under any linear mapping, Exercise 2 gives the following geometric interpretation of an orthogonal matrix: it is rotation around the origin (angles and vector lengths are preserved, while the origin stays in place). Strictly speaking, in case we have rotation and in case - rotation combined with reflection.
Another interpretation is suggested by the next exercise.
Exercise 3. If is an orthonormal basis, then the matrix is orthogonal. Conversely, rows or columns of an orthogonal matrix form an orthonormal basis.
Proof. Orthonormality means that if and if These equations are equivalent to orthogonality of
Exercise 4. Let and be two orthonormal bases. Let be the transition matrix from coordinates in the basis to coordinates in the basis . Then is orthogonal.
Proof. By Exercise 3, both and are orthogonal. Hence, by Exercise 1 is orthogonal. It suffices to show that a product of two orthogonal matrices is orthogonal:
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