Math 21b Orthogonal Matrices T Transpose. The transpose of an n × m matrix A with entries aij is the m × n matrix A with entries aji. The transpose AT and A are related by swapping rows with columns. Properties. Let A, B be matrices and ~v, ~w be vectors. 1.( AB)T = BT AT 2. ~vT ~w = ~v · ~w. 3.( A~v) · ~w = ~v · (AT ~w) 4. If A is invertible, so is AT and (AT )−1 = (A−1)T . 5. ker(AT ) = (im A)? and im(AT ) = (ker A)? Orthogonal Matrices. An invertible matrix A is called orthogonal if A−1 = AT or equivalently, AT A = I. The corresponding linear transformation T (~x) = A~x is called an orthogonal transformation. Caution! Orthogonal projections are almost never orthogonal transformations! Examples. Determine which of the following types of matrices are orthogonal. For those which aren't, under what conditions would they be? cos α − sin α 1. (Rotation) A = sin α cos α Solution. Check if AAT = I: cos α − sin α cos α sin α 1 0 AAT = = sin α cos α − sin α cos α 0 1 so yes this matrix is orthogonal. This shows that any rotation is an orthogonal transforma- tion. a b 2. (Reflection Dilation) B = . b −a Solution. Check if BBT = I: a b a b a2 + b2 0 BBT = = : b −a b −a 0 a2 + b2 This will equal I only if a2 + b2 = 1, i.e., B is a reflection (no dilation). This shows that any reflection is an orthogonal transformation. 2 j j j 3 3 3. (Basis Matrix) S = 4~v1 ~v2 ~v35 where f~v1;~v2;~v3g is a basis for R . j j j Solution. Check if SST = I: 2 3 2 T 3 2 3 j j j | ~v1 | ~v1 · ~v1 ~v1 · ~v2 ~v1 · ~v3 T T SS = 4~v1 ~v2 ~v35 4| ~v2 |5 = 4~v2 · ~v1 ~v2 · ~v2 ~v2 · ~v35 T j j j | ~v3 | ~v3 · ~v1 ~v3 · ~v2 ~v3 · ~v3 The only way this can equal the identity is if ~vi · ~vj = 0 whenever i 6= j and ~vi · ~vi = 1. So, the matrix S is orthogonal if and only if f~v1;~v2;~v3g is an orthonormal basis. Orthogonal Matrices and Bases. An n × n matrix A is orthogonal if and only if its columns form an orthonormal basis for Rn. Caution: This only works if A is square! If A isn't square, then it can't be orthogonal since AAT would have either a row or column of zeros! Orthogonal Matrices and Geometry. Orthonormal matrices preserve dot products, lengths and angles. That is, if A is orthogonal and ~v, ~w are vectors: 1. (Preserves Dot Products) (A~v) · (A~w) = ~v · ~w 2. (Preserves Lengths) kA~vk = k~vk 3. (Preserves Angles) The angle between ~v and ~w is the same as the angle between A~v and A~w. Composition of Orthogonal Transformations. The composition of two orthogonal transformations is again an orthogonal transformation. The inverse of an orthogonal transformation is again orthogonal. Orthogonal Matrices and Projection. If V ⊂ Rn is a linear subspace recall that the orthogonal projec- n n T tion projV : R ! R is given by the matrix A = QQ where the columns of Q is an orthonormal basis for V . Note: The matrix A can't be orthogonal unless V = Rn. (Why?) Examples. Determine if the following transformations are orthogonal. 1. Shear in the plane. Solution. Shears don't preserve lengths, so this can't be an orthogonal transformation (unless we shear by 0). We can check by computing 1 k 1 0 1 + k2 k = 0 1 k 1 k 1 which is not I unless k = 0. (Note, it didn't matter if we considered horizontal or vertical shears because they are just the transpose of eachother!) 2. Projection in three dimensions onto a plane. Solution. Orthogonal projection onto a linear space of smaller dimension is never an orthog- onal transformation. 3. Reflection in two dimensions at the origin. Solution. This is the same as rotation by 180◦ and rotations are orthogonal. So this is an orthogonal transformation. 4. Reflection in three dimensions at a plane. Solution. Reflection preserves lengths and angles, so it is an orthogonal transformation. 5. Dilation with factor of 2. Solution. This transformation changes the lengths of vectors, so isn't orthogonal. 6. Projection in two dimensions onto a plane. Solution. Orthogonal projection is almost never an orthogonal transformation, unless it is the identity projection. This is the identity projection, so it is orthogonal..