Orthogonal Transformations (Matrices)

Orthogonal transformations (Matrices)

Warm-up (a) If M is a (real) matrix whose columns are orthonormal, what can you say about M T M?

(b) If M is a square matrix whose columns are orthonormal, is M invertible? If so, what is its inverse?

A square matrix with orthonormal columns (and real entries) is, rather confusingly, called an orthogonal matrix. There is no special term for a square matrix with orthogonal columns.

1. (T/F) 0 1 1 0 0 0 0 −1 1 0 0 0   1 0 0 0 0 0   0 0 0 0 1 1   0 0 0 0 −1 1 0 0 0 1 0 0 is an orthogonal transformation.

Orthogonal transformation

2. Which of the following is always an orthogonal transformation?

(a) Shears

(b) Projection onto a plane

(d) Reﬂection about a line

(e) Reﬂection about a plane

(f) Dilation

(g) Rotation

(h) Dilation Rotation

(i) The composition of two orthogonal transformations

(j) The inverse of an orthogonal transformation

1 3. If A is an n × m matrix, what is the relationship between rank(AT ) and rank(A)?

4. Suppose M is an invertible matrix. Is M T necessarily invertible? If so, what can you say about its inverse? If not, why not?

5. Suppose A is an n × m matrix and B is an m × p matrix. Decide whether each of the following expressions makes sense; if so, what size is the matrix?

(a)( AB)T (b) AT BT (c) BT AT

6. Complete the following properties:

(a)( AB)T =

(b)( AT )−1 = (assuming A is invertble)

(c)( AT )T =

7. Show that ~x · A~y = AT ~x · ~y

One can use this property to prove that if A is a orthogonal matrix, then A~x · A~y = ~x · ~y. This implies orthogonal transformations preserve the length of vectors and the angles between them.

2 Deﬁnition: A is symmetric if AT = A.

8. (T/F)

(a) If V is any subspace of Rn, then the matrix of proj V is symmetric.

(b) Any symmetric matrix is invertible.

0 3 1 9. Find the orthogonal projection P from 3 to the linear space spanned by ~v = 0 , and ~v = 4 R 1   2 5   1 0

A matrix M whose columns are orthonormal has a special property: M T M is an identity matrix. We are particularly interested in square matrices with orthonormal columns, so these get a special name.

Deﬁnition 1. An orthogonal matrix is a square matrix (with real entries) whose columns are orthonormal.

When M and M T are square matrices, the equation M T M = I tells us that M is invertible, and its inverse is exactly M T . Fact. If M is an orthogonal matrix, then M is invertible, and M −1 = M T .

This fact can save a lot of computational eﬀort when working with orthogonal matrices, since taking a transpose is much simpler than our usually method of calculating the inverse of a matrix.