+1 is a proper rotation, and one with a negative determinant −1 is an improper rotation, that is a reflection combined with a proper rotation. It will now be shown that a proper rotation matrix has at least one invariant vector , i.e., . Because this requires that , we see that the vector must be an eigenvector of the matrix with eigenvalue . Thus, this is equivalent to showing that . Use the two relations for any matrix A and (since ) to compute This shows that is a root (solution) of the characteristic equation, that is, In other words, the matrix is singular and has a non-zero kernel, that is, there is at least one non-zero vector, say , for which The line for real is invariant under , i.e., is a rotation axis. This proves Euler's theorem. Equivalence of an orthogonal matrix to a rotation matrix Two matrices (representing linear maps) are said to be equivalent if there is a change of basis that makes one equal to the other. A proper orthogonal matrix is always equivalent (in this sense) to either the following matrix or to its vertical reflection: Then, any orthogonal matrix is either a rotation or an improper rotation. A general orthogonal matrix has only one real eigenvalue, either +1 or −1. When it is +1 the matrix is a rotation. When −1, the matrix is an improper rotation. If has more than one invariant vector then and . Any vector is an invariant vector of . Excursion into matrix theory In order to prove the previous equation some facts from matrix theory must be recalled. An matrix has orthogonal eigenvectors if and only if is normal, that is, if . This result is equivalent to stating that normal matrices can be brought to diagonal form by a unitary similarity transformation: and is unitary, that is, The eigenvalues are roots of the characteristic equation. If the matrix happens to be unitary (and note that unitary matrices are normal), then and it follows that the eigenvalues of a unitary matrix are on the unit circle in the complex plane: Also an orthogonal (real unitary) matrix has eigenvalues on the unit circle in the complex plane. Moreover, since its characteristic equation (an th order polynomial in ) has real coefficients, it follows that its roots appear in complex conjugate pairs, that is, if is a root