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 p p p 2 3 2 3 2 0 3 1=p18 1= 2 −2=3 p18 p 1. Let U = 44=p18 0p 1=3 5, ~x = 4− 185, and ~y = 4p25. 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 j~xj, jU~xj, j~yj, jU~yj. 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) jU~xj = j~xj.(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 2 p p 3 1=p10 3= p10 0p 2. Find the inverse of the matrix A = 43=p20 −1=p20 −1=p 25. 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.