Let \(M\) be a real nonsingular \(3\times3\) matrix. Prove that there are real matrices \(S\) and \(U\) such that \[ M=SU=US, \] all eigenvalues of \(U\) are equal to \(1\), and \(S\) is diagonalizable over \(\CC\).
Solution
1. If \(M\) has a nonreal eigenvalue, then \(M\) is diagonalizable over\(\CC\).
The characteristic polynomial of the real \(3\times3\) matrix \(M\) is a real cubic. Hence a nonreal eigenvalue \(\lambda\) occurs together with its conjugate \(\overline\lambda\). The remaining root \(\mu\) is real. Thus \[ \lambda,\quad \overline\lambda,\quad \mu \] are three distinct eigenvalues: the first two are distinct and nonreal, whereas \(\mu\) is real. A \(3\times3\) complex matrix with three distinct eigenvalues is diagonalizable over \(\CC\).
2. In the case of step 1, the choice\[ S=M, \qquad U=I \]has all the required properties.
Both matrices are real and \[ SU=US=M. \] Every eigenvalue of \(U=I\) equals \(1\), while step 1 says that \(S=M\) is diagonalizable over \(\CC\).
3. Suppose now that every eigenvalue of \(M\) is real. Then there is areal invertible matrix \(P\) such that\[ J\coloneqq P^{-1}MP \]is block diagonal with real Jordan blocks\[ J_r(\lambda)=\lambda I_r+N_r, \]where \(N_r\) is the nilpotent Jordan block of size \(r\) and every\(\lambda\ne0\).
Since the characteristic polynomial splits over \(\RR\) in this case, the Jordan normal form theorem can be applied over \(\RR\), so the Jordan basis may be chosen real. Nonsingularity of \(M\) means that \(0\) is not an eigenvalue, hence every block eigenvalue \(\lambda\) is nonzero.
4. For each Jordan block from step 3, define\[ S_r\coloneqq\lambda I_r, \qquad U_r\coloneqq I_r+\lambda^{-1}N_r. \]Then\[ J_r(\lambda)=S_rU_r=U_rS_r, \]\(S_r\) is diagonal, and every eigenvalue of \(U_r\) equals \(1\).
Because \(S_r\) is scalar, \[ \begin{aligned} S_rU_r &= \lambda I_r \left(I_r+\lambda^{-1}N_r\right)\\ &= \lambda I_r+N_r = J_r(\lambda), \end{aligned} \] and the same computation gives \(U_rS_r=J_r(\lambda)\). The matrix \(S_r\) is already diagonal. The matrix \(U_r\) is upper triangular with every diagonal entry equal to \(1\), so all its eigenvalues equal \(1\).
5. Let \(S_0\) and \(U_0\) be the block-diagonal matrices obtained byplacing the matrices \(S_r\) and \(U_r\) from step 4 in the same blockpositions as the Jordan blocks of \(J\). Then\[ J=S_0U_0=U_0S_0, \]\(S_0\) is diagonal, and every eigenvalue of \(U_0\) equals \(1\).
4. A block-diagonalmatrix whose blocks are diagonal scalar matrices is diagonal, and theeigenvalues of a block-diagonal matrix are the eigenvalues of its blocks.
6. Define\[ S\coloneqq PS_0P^{-1}, \qquad U\coloneqq PU_0P^{-1}. \]Then \(S\) and \(U\) are real and satisfy all the required properties.
The matrices \(P\), \(S_0\), and \(U_0\) are real, so \(S\) and \(U\) are real. Using step 5, \[ \begin{aligned} SU &= P(S_0U_0)P^{-1} = PJP^{-1} = M,\\ US &= P(U_0S_0)P^{-1} = PJP^{-1} = M. \end{aligned} \] Similarity preserves eigenvalues, so every eigenvalue of \(U\) equals \(1\). Also \(S\) is similar over \(\RR\) to the diagonal matrix \(S_0\), hence is diagonalizable over \(\RR\) and therefore over \(\CC\).
7. Q.E.D.
If \(M\) has a nonreal eigenvalue, step 2 gives the required matrices. If all eigenvalues are real, step 6 gives them. These cases exhaust all real \(3\times3\) matrices.