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) 20 1 1 0 0 03 60 −1 1 0 0 07 6 7 61 0 0 0 0 07 6 7 60 0 0 0 1 17 6 7 40 0 0 0 −1 15 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 (c) Projection onto a line (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. 203 233 1 9. Find the orthogonal projection P from 3 to the linear space spanned by ~v = 0 , and ~v = 4 R 1 4 5 2 5 4 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. 3.