Theorem. If V is a finite-dimensional vector space with given inner product, f an orthogonal linear map from V to V, U a linear subspace of V, and if f carries each vector u in U to a vector f(u) in U, then f carries each vector in the orthogonal complement U^{perp} of U to a vector in U^{perp}.
The preceding theorem means that the task of understanding the geometric structure of an orthogonal linear map is simplified when that map admits an ``invariant subspace''. In that case one only needs to understand what the map does to the subspace and to its orthogonal complement.
Definition. If V is a vector space with given inner product I, then a linear map f from V to itself is said to be a symmetric linear map if for all v, w in V one has I(f(v), w) = I(v, f(w)).
Proposition. If V is a finite-dimensional vector space with given inner product and f a linear map from V to V, then f is a symmetric linear map if and only if its matrix relative to any orthonormal basis of V is a symmetric matrix.
Theorem. If V is a finite-dimensional vector space with given inner product, f a symmetric linear map from V to V, U a linear subspace of V, and if f carries each vector u in U to a vector f(u) in U, then f carries each vector in the orthogonal complement U^{perp} of U to a vector in U^{perp}.
Let f be the linear map from R^{2} to itself given by f(x) = M x where M is the matrix
| 3/5 | 4/5 |
| 4/5 | -3/5 |
Explain why f is both an orthogonal linear map and a symmetric linear map. What linear subspaces of R^{2} are carried to themselves by f ?
Let f be the rotation of R^{3} about its third coordinate axis through the angle theta. What is the matrix of f relative to the standard basis of R^{3} ?
Must the product of two symmetric matrices be a symmetric matrix?
Show that the sum of two symmetric matrices is a symmetric matrix.
Must the sum of two orthogonal matrices be an orthogonal matrix?
Let f be the linear map from R^{2} to itself given by f(x) = M x where M is the matrix
| 3/5 | -4/5 |
| 4/5 | 3/5 |
Show that no 1-dimensional linear subspace of R^{2} is carried to itself by f.