Orthogonal matrices
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
(1)
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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