Orthogonal Matrices and Transformations

First, recall the transpose of a matrix...

The Transpose of a Matrix Let A be an m × n matrix. The transpose of A, denoted by AT , is the n × m matrix such that the jth column of A is the jth row of AT . (Equivalently, the ith row of A is the ith column of AT .) In other words, we find AT by taking the columns of A (in order) and making them the rows of AT (in order). Properties of the Transpose

1)( AT )T = A

2)( AB)T = BT AT

√ √ √      0  1/√18 1/ 2 −2/3 √18 √ 1. Let U = 4/√18 0√ 1/3 , ~x = − 18, and ~y = √2. 1/ 18 −1/ 2 −2/3 0 2

(a) Verify that the column vectors of A are orthonormal, and thus form an orthonormal basis for R3. (No Gram-Schmidt necessary!)

(b) Find the matrix product U T U.

(c) Find the vectors U~x and U~y.

1 (d) Compute |~x|, |U~x|, |~y|, |U~y|. What do you notice?

(e) Find the angle between the vectors ~x and ~y. Then find the angle between the vectors U~x and U~y. What do you notice?

Orthogonal Matrices/Transformations An orthogonal matrix is an n × n (square) matrix U such that:

U T U = I

Equivalently, an orthogonal matrix is an n × n (square) matrix U such that:

U −1 = U T

We say that a linear transformation is orthogonal if it can be represented by an orthogonal matrix.

Properties of Orthogonal Matrices

Suppose U is an n × n orthogonal matrix. Then for all ~x and ~y in Rn:

1) |U~x| = |~x|.(U preserves the length of vectors.)

2) U~x · U~y = ~x · ~y.(U preserves the angle between two vectors, by preserving the dot product.)

3) The columns of U form an orthonormal basis for Rn. (In fact, this works both ways. If the columns of a square matrix are orthonormal, then the matrix itself must be orthogonal. Can you show why?)

Warning: The columns of an orthogonal matrix must be orthonormal! If the columns of a matrix are only orthogonal (and not unit vectors), then the matrix would NOT be orthogonal! For example, 2 3 A = has orthogonal columns, but A is not an orthogonal matrix. 3 −2

2  √ √  1/√10 3/ √10 0√ 2. Find the inverse of the matrix A = 3/√20 −1/√20 −1/√ 2. Check your answer via matrix 3/ 20 −1/ 20 1/ 2 multiplication.

3. Which of the following types of transformations can never be an orthogonal transformation? Which of them must be an orthogonal transformation?

• The Identity Transformation

• Dilations

• Reflections

• Rotations

• Projections

• Shears

4. Determine if the following statements are true or false. If they are true, give a justification. If they are false, give a counterexample.

(a) If a matrix is orthogonal, then it must be invertible.

(b) If a matrix is invertible, then it must be orthogonal.

(c) If A is a 2 × 3 matrix and AT A = I, then A must have orthonormal columns.

3 (d) If A is an orthogonal matrix, then AT is also an orthogonal matrix.

(e) If A is an orthogonal matrix, then A−1 is also an orthogonal matrix.

(f) If A is an orthogonal matrix, then A2 is also an orthogonal matrix.

(g) If A and B are orthogonal n × n matrices, then AB is orthogonal.

(h) If A and B are orthogonal n × n matrices, then A + B is orthogonal.

(i) If the column vectors of A are orthonormal, then the row vectors of A must also be orthonormal. (Hint: Use part (d).)

4