Definition. Two square matrices A and B of the same size are said to be similar if there is an invertible matrix U such that B = U A U^{-1}. If the relation B = U A U^{-1} holds, then B is called the conjugate of A by U.
Two matrices that represent a linear transformation of a finite-dimensional vector space in different bases are always similar matrices. In this case the conjugating matrix is the ``matrix of basis change''. (It can be difficult to avoid confusing the matrix of basis change with its inverse.)
Theorem. A linear transformation of a finite-dimensional vector space is represented by a diagonal matrix, relative to a given basis of the vector space, if and only if each vector in the given basis is a characteristic vector, i.e., each vector in the given basis generates a subspace that is a 1-dimensional characteristic subspace.
Corollary. A square matrix M is similar to a diagonal matrix if and only if there is an invertible matrix V with the property that for each column V_{j} of V the column M V_{j} is parallel to V_{j}. This is the case for M and V if and only if V^{-1} M V is a diagonal matrix.
Let f be the linear transformation of R^{3} that is given by the matrix
| 0 | 1 | 0 |
| 0 | 0 | 1 |
| 1 | 0 | 0 |
Show that f is an orthogonal linear transformation.
Decompose R^{3} into mutually orthogonal f-invariant subspaces of minimal dimensions.
Find an orthonormal basis of each of the f-invariant subspaces.
Explain why the union of the orthonormal bases of the subspaces is an orthonormal basis of R^{3}.
Find the matrix of f relative to the orthormal basis of the previous part.
Let f be the linear transformation that is defined by f(x) = M x where M is the 4 \times 4 matrix
| 0 | 1 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 1 |
| 1 | 0 | 0 | 0 |
Show that f is an orthogonal linear transformation.
Show that f admits an invariant linear subspace of dimension 2.
What is the orthogonal complement of the subspace in the previous part?