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

a b  2. (Reflection Dilation) B = . b −a

 | | |  3 3. (Basis Matrix) S = ~v1 ~v2 ~v3 where {~v1,~v2,~v3} is a basis for R . | | | 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.

2. Projection in three dimensions onto a plane.

3. Reflection in two dimensions at the origin.

4. Reflection in three dimensions at a plane.

5. Dilation with factor of 2.

6. Projection in two dimensions onto a plane.